domain listlengths 1 3 | difficulty float64 1.5 9.5 | problem stringlengths 85 790 | solution stringlengths 6 4.89k | answer stringlengths 1 139 | source stringclasses 23
values |
|---|---|---|---|---|---|
[
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 8 | Find a real number $t$ such that for any set of 120 points $P_1, \ldots P_{120}$ on the boundary of a unit square, there exists a point $Q$ on this boundary with $|P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t$. |
We need to find a real number \( t \) such that for any set of 120 points \( P_1, \ldots, P_{120} \) on the boundary of a unit square, there exists a point \( Q \) on this boundary with \( |P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t \).
Define \(\mathcal{U}\) to be a set of points \( P_1, \ldots, P_{120} \) on the boun... | 30(1 + \sqrt{5}) | usa_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangles -> Other",
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 6.5 | Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB \equal{} \angle CQB \equal{} 45^\circ$, $ \angle ABP \equal{} \angle QBC \equal{} 75^\circ$, $ \angle RAC \equal{} 105^\circ$, and $ RQ^2 \equal{} 6CM^2$, ... |
Let \( ABP, BCQ, CAR \) be three non-overlapping triangles erected outside of acute triangle \( ABC \). Let \( M \) be the midpoint of segment \( AP \). Given that \( \angle PAB = \angle CQB = 45^\circ \), \( \angle ABP = \angle QBC = 75^\circ \), \( \angle RAC = 105^\circ \), and \( RQ^2 = 6CM^2 \), we aim to compute... | \frac{2}{3} | usa_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Discrete Mathematics -> Logic"
] | 8 | Let $S$ be a set, $|S|=35$. A set $F$ of mappings from $S$ to itself is called to be satisfying property $P(k)$, if for any $x,y\in S$, there exist $f_1, \cdots, f_k \in F$ (not necessarily different), such that $f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y))))$.
Find the least positive integer $m$, such tha... |
Let \( S \) be a set with \( |S| = 35 \). A set \( F \) of mappings from \( S \) to itself is said to satisfy property \( P(k) \) if for any \( x, y \in S \), there exist \( f_1, f_2, \ldots, f_k \in F \) (not necessarily different) such that \( f_k(f_{k-1}(\cdots (f_1(x)) \cdots )) = f_k(f_{k-1}(\cdots (f_1(y)) \cdot... | 595 | china_national_olympiad |
[
"Mathematics -> Number Theory -> Factorization"
] | 7 | Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form \[\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},\]
where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$ |
Let \( n \) be a positive integer. We aim to find the least positive integer \( d_n \) which cannot be expressed in the form
\[
\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},
\]
where \( a_i \) and \( b_i \) are nonnegative integers for each \( i \).
We claim that the minimal number that is not \( n \)-good is
\[
d_n = 2 \le... | 2 \left( \frac{4^n - 1}{3} \right) + 1 | usa_team_selection_test |
[
"Mathematics -> Number Theory -> Congruences"
] | 7.5 | Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:
i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime
ii) $\sum^n_{i=1} a_... |
Let \( m > 1 \) be an integer. We are tasked with finding the smallest positive integer \( n \) such that for any integers \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \), there exist integers \( x_1, x_2, \ldots, x_n \) satisfying the following two conditions:
1. There exists \( i \in \{1, 2, \ldots, n\}... | 2\omega(m) + 1 | china_national_olympiad |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 7 | Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter o... |
Given a convex quadrilateral \(ABCD\) inscribed in a circle with \(\angle A = 60^\circ\), \(BC = CD = 1\), and the intersections of rays \(AB\) and \(DC\) at point \(E\), and rays \(BC\) and \(AD\) at point \(F\), we aim to find the perimeter of quadrilateral \(ABCD\) given that the perimeters of triangles \(BCE\) and... | \frac{38}{7} | china_team_selection_test |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 8 | Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression
\[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^... |
Given two integers \( m, n \) which are greater than 1, and two positive real numbers \( r, s \) such that \( r < s \), we aim to find the maximal value of the expression
\[
f = \frac{\left( \sum_{j=1}^{n} \left( \sum_{i=1}^{m} a_{ij}^s \right)^{\frac{r}{s}} \right)^{\frac{1}{r}}}{\left( \sum_{i=1}^{m} \sum_{j=1}^{n} ... | \min(m, n)^{\frac{1}{r} - \frac{1}{s}} | china_team_selection_test |
[
"Mathematics -> Number Theory -> Congruences"
] | 8 | Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely. |
Given two positive integers \( p \) and \( q \), we are to determine the smallest number \( n \) such that the operation of choosing two identical numbers \( a, a \) on the blackboard and replacing them with \( a+p \) and \( a+q \) can go on infinitely.
To solve this, we first note that we can assume \(\gcd(p, q) = 1... | \frac{p+q}{\gcd(p,q)} | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 8 | Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have
\[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}}
\le \lambda\]
Where $a_{n+i}=a_i,i=1,2,\ldots,... |
Given positive integers \( n \) and \( k \) such that \( n \geq 4k \), we aim to find the minimal value \( \lambda = \lambda(n, k) \) such that for any positive reals \( a_1, a_2, \ldots, a_n \), the following inequality holds:
\[
\sum_{i=1}^{n} \frac{a_i}{\sqrt{a_i^2 + a_{i+1}^2 + \cdots + a_{i+k}^2}} \leq \lambda,
\... | n - k | china_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5.5 | Let $P_1P_2\ldots P_{24}$ be a regular $24$-sided polygon inscribed in a circle $\omega$ with circumference $24$. Determine the number of ways to choose sets of eight distinct vertices from these $24$ such that none of the arcs has length $3$ or $8$. |
Let \( P_1P_2\ldots P_{24} \) be a regular 24-sided polygon inscribed in a circle \(\omega\) with circumference 24. We aim to determine the number of ways to choose sets of eight distinct vertices from these 24 such that none of the arcs has length 3 or 8.
We generalize the problem by considering a regular polygon wi... | 258 | china_national_olympiad |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 8 | Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely ma... |
### Part 1:
We need to find the smallest integer \( k \) such that if the minimum number in \( A, B \in S \) are both \(\geq k\), then \( A \rightarrow B \) implies \( B \rightarrow A \).
We claim that the smallest integer \( k \) is \( 8 \).
**Proof:**
1. **\( k \leq 7 \) does not satisfy the condition:**
Con... | 10^8 | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 5 | Let $S$ be the set of all points in the plane whose coordinates are positive integers less than or equal to 100 (so $S$ has $100^{2}$ elements), and let $\mathcal{L}$ be the set of all lines $\ell$ such that $\ell$ passes through at least two points in $S$. Find, with proof, the largest integer $N \geq 2$ for which it ... | Let the lines all have slope $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Without loss of generality, let this slope be positive. Consider the set of points that consists of the point of $S$ with the smallest coordinates on each individual line in the set $L$. Consider a point $(x, y)$ in this, because there i... | 4950 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 8 | Consider an $n$ -by- $n$ board of unit squares for some odd positive integer $n$ . We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ ... | We claim the answer is $(\frac{n+1}{2})^2$ .
First, consider a checkerboard tiling of the board with 4 colors: R, G, B, Y. Number each column from $1$ to $n$ from left to right and each row from $1$ to $n$ from top to bottom. We color a tile R if its row and column are odd, a tile G is its row is even but its column is... | \[
\left(\frac{n+1}{2}\right)^2
\] | usamo |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 3.5 | Six students taking a test sit in a row of seats with aisles only on the two sides of the row. If they finish the test at random times, what is the probability that some student will have to pass by another student to get to an aisle? | The probability $p$ that no student will have to pass by another student to get to an aisle is the probability that the first student to leave is one of the students on the end, the next student to leave is on one of the ends of the remaining students, etc.: $p=\frac{2}{6} \cdot \frac{2}{5} \cdot \frac{2}{4} \cdot \fra... | \frac{43}{45} | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 4 | Some people like to write with larger pencils than others. Ed, for instance, likes to write with the longest pencils he can find. However, the halls of MIT are of limited height $L$ and width $L$. What is the longest pencil Ed can bring through the halls so that he can negotiate a square turn? | $3 L$. | 3 L | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Graph Theory",
"Mathematics -> Algebra -> Abstract Algebra -> Other (Recurrence Relations) -> Other",
"Mathematics -> Algebra -> Other (Number Theory - Divisibility) -> Other"
] | 5 | Let $S$ be the set of $3^{4}$ points in four-dimensional space where each coordinate is in $\{-1,0,1\}$. Let $N$ be the number of sequences of points $P_{1}, P_{2}, \ldots, P_{2020}$ in $S$ such that $P_{i} P_{i+1}=2$ for all $1 \leq i \leq 2020$ and $P_{1}=(0,0,0,0)$. (Here $P_{2021}=P_{1}$.) Find the largest integer ... | From $(0,0,0,0)$ we have to go to $( \pm 1, \pm 1, \pm 1, \pm 1)$, and from $(1,1,1,1)$ (or any of the other similar points), we have to go to $(0,0,0,0)$ or $(-1,1,1,1)$ and its cyclic shifts. If $a_{i}$ is the number of ways to go from $(1,1,1,1)$ to point of the form $( \pm 1, \pm 1, \pm 1, \pm 1)$ in $i$ steps, the... | 4041 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 6 | Anastasia is taking a walk in the plane, starting from $(1,0)$. Each second, if she is at $(x, y)$, she moves to one of the points $(x-1, y),(x+1, y),(x, y-1)$, and $(x, y+1)$, each with $\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$. What is the probability that $k$ is divisible ... | The key idea is to consider $(a+b, a-b)$, where $(a, b)$ is where Anastasia walks on. Then, the first and second coordinates are independent random walks starting at 1, and we want to find the probability that the first is divisible by 3 when the second reaches 0 for the first time. Let $C_{n}$ be the $n$th Catalan num... | \frac{3-\sqrt{3}}{3} | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 5.25 | Let $\triangle A B C$ be a triangle inscribed in a unit circle with center $O$. Let $I$ be the incenter of $\triangle A B C$, and let $D$ be the intersection of $B C$ and the angle bisector of $\angle B A C$. Suppose that the circumcircle of $\triangle A D O$ intersects $B C$ again at a point $E$ such that $E$ lies on ... | Consider the following lemma: Lemma. $A D \perp E O$. Proof. By the Shooting Lemma, the reflection of the midpoint $M$ of arc $B C$ not containing $A$ over $B C$ lies on $(A D O)$. Hence $\measuredangle A D E+\measuredangle D E O=\measuredangle M D C+\measuredangle D M^{\prime} O=\measuredangle M D C+\measuredangle M^{... | \frac{15}{169} | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 4.5 | Let $W, S$ be as in problem 32. Let $A$ be the least positive integer such that an acute triangle with side lengths $S, A$, and $W$ exists. Find $A$. | There are two solutions to the alphametic in problem 32: $36 \times 686=24696$ and $86 \times 636=54696$. So $(W, S)$ may be $(3,2)$ or $(8,5)$. If $(W, S)=(3,2)$, then by problem (3) $A=3$, but then by problem $31 W=4$, a contradiction. So, $(W, S)$ must be $(8,5)$. By problem $33, A=7$, and this indeed checks in prob... | 7 | HMMT_2 |
[
"Mathematics -> Number Theory -> Other (since the context of \\( A \\) is necessary but unspecified here, the question relates to determining and summing all divisors of an integer) -> Other"
] | 4.5 | Let $A$ be as in problem 33. Let $W$ be the sum of all positive integers that divide $A$. Find $W$. | Problems 31-33 go together. See below. | 8 | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 5 | Point $P$ is inside a square $A B C D$ such that $\angle A P B=135^{\circ}, P C=12$, and $P D=15$. Compute the area of this square. | Let $x=A P$ and $y=B P$. Rotate $\triangle B A P$ by $90^{\circ}$ around $B$ to get $\triangle B C Q$. Then, $\triangle B P Q$ is rightisosceles, and from $\angle B Q C=135^{\circ}$, we get $\angle P Q C=90^{\circ}$. Therefore, by Pythagorean's theorem, $P C^{2}=x^{2}+2y^{2}$. Similarly, $P D^{2}=y^{2}+2x^{2}$. Thus, $... | 123+6\sqrt{119} | HMMT_2 |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 7 | For each prime $p$, a polynomial $P(x)$ with rational coefficients is called $p$-good if and only if there exist three integers $a, b$, and $c$ such that $0 \leq a<b<c<\frac{p}{3}$ and $p$ divides all the numerators of $P(a)$, $P(b)$, and $P(c)$, when written in simplest form. Compute the number of ordered pairs $(r, s... | By Vieta, the sum of the roots is $-10(\bmod p)$. However, since the three roots are less than $p/3$, it follows that the roots are $\left(p-a^{\prime}\right)/3,\left(p-b^{\prime}\right)/3,\left(p-c^{\prime}\right)/3$, where there are finitely many choices $a^{\prime}<b^{\prime}<c^{\prime}$. By pigeonhole, one choice, ... | 12 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Consider the cube whose vertices are the eight points $(x, y, z)$ for which each of $x, y$, and $z$ is either 0 or 1 . How many ways are there to color its vertices black or white such that, for any vertex, if all of its neighbors are the same color then it is also that color? Two vertices are neighbors if they are the... | Divide the 8 vertices of the cube into two sets $A$ and $B$ such that each set contains 4 vertices, any two of which are diagonally adjacent across a face of the cube. We do casework based on the number of vertices of each color in set $A$. - Case 1: 4 black. Then all the vertices in $B$ must be black, for 1 possible c... | 118 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4.5 | How many elements are in the set obtained by transforming $\{(0,0),(2,0)\} 14$ times? | Transforming it $k \geq 1$ times yields the diamond $\{(n, m):|n-1|+|m| \leq k+1\}$ with the points $(1, k),(1, k+1),(1,-k),(1,-k-1)$ removed (this can be seen inductively). So we get $(k+1)^{2}+k^{2}-4$ lattice points, making the answer 477. | 477 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 4.5 | A deck of 100 cards is labeled $1,2, \ldots, 100$ from top to bottom. The top two cards are drawn; one of them is discarded at random, and the other is inserted back at the bottom of the deck. This process is repeated until only one card remains in the deck. Compute the expected value of the label of the remaining card... | Note that we can just take averages: every time you draw one of two cards, the EV of the resulting card is the average of the EVs of the two cards. This average must be of the form $$2^{\bullet} \cdot 1+2^{\bullet} \cdot 2+2^{\bullet} \cdot 3+\cdots+2^{\bullet} \cdot 100$$ where the $2^{\bullet}$ add up to 1. Clearly, ... | \frac{467}{8} | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 4 | Points $X$ and $Y$ are inside a unit square. The score of a vertex of the square is the minimum distance from that vertex to $X$ or $Y$. What is the minimum possible sum of the scores of the vertices of the square? | Let the square be $A B C D$. First, suppose that all four vertices are closer to $X$ than $Y$. Then, by the triangle inequality, the sum of the scores is $A X+B X+C X+D X \geq A B+C D=2$. Similarly, suppose exactly two vertices are closer to $X$ than $Y$. Here, we have two distinct cases: the vertices closer to $X$ are... | \frac{\sqrt{6}+\sqrt{2}}{2} | HMMT_2 |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5.25 | Over all pairs of complex numbers $(x, y)$ satisfying the equations $$x+2y^{2}=x^{4} \quad \text{and} \quad y+2x^{2}=y^{4}$$ compute the minimum possible real part of $x$. | Note the following observations: (a) if $(x, y)$ is a solution then $(\omega x, \omega^{2} y)$ is also a solution if $\omega^{3}=1$ and $\omega \neq 1$. (b) we have some solutions $(x, x)$ where $x$ is a solution of $x^{4}-2x^{2}-x=0$. These are really the only necessary observations and the first does not need to be n... | \sqrt[3]{\frac{1-\sqrt{33}}{2}} | HMMT_2 |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 5.25 | Compute the number of complex numbers $z$ with $|z|=1$ that satisfy $$1+z^{5}+z^{10}+z^{15}+z^{18}+z^{21}+z^{24}+z^{27}=0$$ | Let the polynomial be $f(z)$. One can observe that $$f(z)=\frac{1-z^{15}}{1-z^{5}}+z^{15} \frac{1-z^{15}}{1-z^{3}}=\frac{1-z^{20}}{1-z^{5}}+z^{18} \frac{1-z^{12}}{1-z^{3}}$$ so all primitive 15th roots of unity are roots, along with -1 and $\pm i$. To show that there are no more, we can try to find $\operatorname{gcd}(... | 11 | HMMT_2 |
[
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 4.5 | Let $E$ be a three-dimensional ellipsoid. For a plane $p$, let $E(p)$ be the projection of $E$ onto the plane $p$. The minimum and maximum areas of $E(p)$ are $9 \pi$ and $25 \pi$, and there exists a $p$ where $E(p)$ is a circle of area $16 \pi$. If $V$ is the volume of $E$, compute $V / \pi$. | Let the three radii of $E$ be $a<b<c$. We know that $ab=9$ and $bc=25$. Consider the plane $p$ where projection $E(p)$ has area $9 \pi$. Fixing $p$, rotate $E$ on the axis passing through the radius with length $b$ until $E(p)$ has area $25 \pi$. The projection onto $p$ will be an ellipse with radii $b$ and $r$, where ... | 75 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5.25 | Compute the number of labelings $f:\{0,1\}^{3} \rightarrow\{0,1, \ldots, 7\}$ of the vertices of the unit cube such that $$\left|f\left(v_{i}\right)-f\left(v_{j}\right)\right| \geq d\left(v_{i}, v_{j}\right)^{2}$$ for all vertices $v_{i}, v_{j}$ of the unit cube, where $d\left(v_{i}, v_{j}\right)$ denotes the Euclidean... | Let $B=\{0,1\}^{3}$, let $E=\{(x, y, z) \in B: x+y+z$ is even $\}$, and let $O=\{(x, y, z) \in B$ : $x+y+z$ is odd $\}$. As all pairs of vertices within $E$ (and within $O$ ) are $\sqrt{2}$ apart, is easy to see that $\{f(E), f(O)\}=\{\{0,2,4,6\},\{1,3,5,7\}\}$. - There are two ways to choose $f(E)$ and $f(O)$; from no... | 144 | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Area",
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 5 | Let $P$ be the set of points $$\{(x, y) \mid 0 \leq x, y \leq 25, x, y \in \mathbb{Z}\}$$ and let $T$ be the set of triangles formed by picking three distinct points in $P$ (rotations, reflections, and translations count as distinct triangles). Compute the number of triangles in $T$ that have area larger than 300. | Lemma: The area of any triangle inscribed in an $a$ by $b$ rectangle is at most $\frac{ab}{2}$. (Any triangle's area can be increased by moving one of its sides to a side of the rectangle). Given this, because any triangle in $T$ is inscribed in a $25 \times 25$ square, we know that the largest possible area of a trian... | 436 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4.25 | The L shape made by adjoining three congruent squares can be subdivided into four smaller L shapes. Each of these can in turn be subdivided, and so forth. If we perform 2005 successive subdivisions, how many of the $4^{2005}$ L's left at the end will be in the same orientation as the original one? | After $n$ successive subdivisions, let $a_{n}$ be the number of small L's in the same orientation as the original one; let $b_{n}$ be the number of small L's that have this orientation rotated counterclockwise $90^{\circ}$; let $c_{n}$ be the number of small L's that are rotated $180^{\circ}$; and let $d_{n}$ be the nu... | 4^{2004}+2^{2004} | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Five people are at a party. Each pair of them are friends, enemies, or frenemies (which is equivalent to being both friends and enemies). It is known that given any three people $A, B, C$ : - If $A$ and $B$ are friends and $B$ and $C$ are friends, then $A$ and $C$ are friends; - If $A$ and $B$ are enemies and $B$ and $... | If $A$ and $B$ are frenemies, then regardless of whether another person $C$ is friends or enemies with $A$, $C$ will have to be frenemies with $B$ and vice versa. Therefore, if there is one pair of frenemies then all of them are frenemies with each other, and there is only one possibility. If there are no frenemies, th... | 17 | HMMT_2 |
[
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 4 | Two vertices of a cube are given in space. The locus of points that could be a third vertex of the cube is the union of $n$ circles. Find $n$. | Let the distance between the two given vertices be 1. If the two given vertices are adjacent, then the other vertices lie on four circles, two of radius 1 and two of radius $\sqrt{2}$. If the two vertices are separated by a diagonal of a face of the cube, then the locus of possible vertices adjacent to both of them is ... | 10 | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 6 | $A B C$ is an acute triangle with incircle $\omega$. $\omega$ is tangent to sides $\overline{B C}, \overline{C A}$, and $\overline{A B}$ at $D, E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\Gamma$, the circle with diameter $\overline{A P}$, is tangent to $\omega$. $\Gamma$ intersects $\o... | By the Law of Sines we have $\sin \angle A=\frac{X Y}{A P}=\frac{4}{5}$. Let $I, T$, and $Q$ denote the center of $\omega$, the point of tangency between $\omega$ and $\Gamma$, and the center of $\Gamma$ respectively. Since we are told $A B C$ is acute, we can compute $\tan \angle \frac{A}{2}=\frac{1}{2}$. Since $\angl... | \frac{675}{4} | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Area",
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 5.25 | Find the smallest possible area of an ellipse passing through $(2,0),(0,3),(0,7)$, and $(6,0)$. | Let $\Gamma$ be an ellipse passing through $A=(2,0), B=(0,3), C=(0,7), D=(6,0)$, and let $P=(0,0)$ be the intersection of $A D$ and $B C$. $\frac{\text { Area of } \Gamma}{\text { Area of } A B C D}$ is unchanged under an affine transformation, so we just have to minimize this quantity over situations where $\Gamma$ is... | \frac{56 \pi \sqrt{3}}{9} | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4.5 | You start out with a big pile of $3^{2004}$ cards, with the numbers $1,2,3, \ldots, 3^{2004}$ written on them. You arrange the cards into groups of three any way you like; from each group, you keep the card with the largest number and discard the other two. You now again arrange these $3^{2003}$ remaining cards into gr... | We claim that if you have cards numbered $1,2, \ldots, 3^{2 n}$ and perform $2 n$ successive grouping operations, then $c$ is a possible value for your last remaining card if and only if $$3^{n} \leq c \leq 3^{2 n}-3^{n}+1$$ This gives $3^{2 n}-2 \cdot 3^{n}+2$ possible values of $c$, for a final answer of $3^{2004}-2 ... | 3^{2004}-2 \cdot 3^{1002}+2 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Number Theory -> Factorization"
] | 6 | The squares of a $3 \times 3$ grid are filled with positive integers such that 1 is the label of the upperleftmost square, 2009 is the label of the lower-rightmost square, and the label of each square divides the one directly to the right of it and the one directly below it. How many such labelings are possible? | We factor 2009 as $7^{2} \cdot 41$ and place the 41 's and the 7 's in the squares separately. The number of ways to fill the grid with 1's and 41 's so that the divisibility property is satisfied is equal to the number of nondecreasing sequences $a_{1}, a_{2}, a_{3}$ where each $a_{i} \in\{0,1,2,3\}$ and the sequence ... | 2448 | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 6 | Let $\Delta A_{1} B_{1} C$ be a triangle with $\angle A_{1} B_{1} C=90^{\circ}$ and $\frac{C A_{1}}{C B_{1}}=\sqrt{5}+2$. For any $i \geq 2$, define $A_{i}$ to be the point on the line $A_{1} C$ such that $A_{i} B_{i-1} \perp A_{1} C$ and define $B_{i}$ to be the point on the line $B_{1} C$ such that $A_{i} B_{i} \perp... | We claim that $\Gamma_{2}$ is the incircle of $\triangle B_{1} A_{2} C$. This is because $\triangle B_{1} A_{2} C$ is similar to $A_{1} B_{1} C$ with dilation factor $\sqrt{5}-2$, and by simple trigonometry, one can prove that $\Gamma_{2}$ is similar to $\Gamma_{1}$ with the same dilation factor. By similarities, we ca... | 4030 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Probability -> Other",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4.5 | Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right 1 unit with equal probability at each step.) If she lands on a point of the form $(6 m, 6 n)$ for $m, n \in \mathbb{Z}$, she ascends to heaven, but if she lands on a point of the for... | Let $P(m, n)$ be the probability that she ascends to heaven from point $(m, n)$. Then $P(6 m, 6 n)=1$ and $P(6 m+3,6 n+3)=0$ for all integers $m, n \in \mathbb{Z}$. At all other points, $$\begin{equation*} 4 P(m, n)=P(m-1, n)+P(m+1, n)+P(m, n-1)+P(m, n+1) \tag{1} \end{equation*}$$ This gives an infinite system of equat... | \frac{13}{22} | HMMT_2 |
[
"Mathematics -> Algebra -> Abstract Algebra -> Field Theory"
] | 6 | Let $p>2$ be a prime number. $\mathbb{F}_{p}[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_{p}$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ are equal in $\mathbb{F}_{p}$ for each nonneg... | Answer: $4 p(p-1)$ Solution 1. First, notice that $(\operatorname{deg} f)(\operatorname{deg} g)=p^{2}$ and both polynomials are clearly nonconstant. Therefore there are three possibilities for the ordered pair $(\operatorname{deg} f, \operatorname{deg} g)$, which are $\left(1, p^{2}\right),\left(p^{2}, 1\right)$, and $... | 4 p(p-1) | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 4 | It is known that exactly one of the three (distinguishable) musketeers stole the truffles. Each musketeer makes one statement, in which he either claims that one of the three is guilty, or claims that one of the three is innocent. It is possible for two or more of the musketeers to make the same statement. After hearin... | We divide into cases, based on the number of distinct people that statements are made about. - The statements are made about 3 distinct people. Then, since exactly one person is guilty, and because exactly one of the three lied, there are either zero statements of guilt or two statements of guilt possible; in either ca... | 99 | HMMT_2 |
[
"Mathematics -> Number Theory -> Prime Numbers"
] | 5.25 | Let $f(n)$ be the largest prime factor of $n^{2}+1$. Compute the least positive integer $n$ such that $f(f(n))=n$. | Suppose $f(f(n))=n$, and let $m=f(n)$. Note that we have $mn \mid m^{2}+n^{2}+1$. First we find all pairs of positive integers that satisfy this condition, using Vieta root jumping. Suppose $m^{2}+n^{2}+1=kmn$, for some positive integer $k$. Considering this as a quadratic in $m$, let the other root (besides $m$) be $m... | 89 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | What is the maximum number of bishops that can be placed on an $8 \times 8$ chessboard such that at most three bishops lie on any diagonal? | If the chessboard is colored black and white as usual, then any diagonal is a solid color, so we may consider bishops on black and white squares separately. In one direction, the lengths of the black diagonals are $2,4,6,8,6,4$, and 2 . Each of these can have at most three bishops, except the first and last which can h... | 38 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 4.5 | Manya has a stack of $85=1+4+16+64$ blocks comprised of 4 layers (the $k$ th layer from the top has $4^{k-1}$ blocks). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top ... | Each time Laura removes a block, 4 additional blocks are exposed, increasing the total number of exposed blocks by 3 . She removes 5 blocks, for a total of $1 \cdot 4 \cdot 7 \cdot 10 \cdot 13$ ways. However, the stack originally only has 4 layers, so we must subtract the cases where removing a block on the bottom laye... | 3384 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4 | Let $X$ be the collection of all functions $f:\{0,1, \ldots, 2016\} \rightarrow\{0,1, \ldots, 2016\}$. Compute the number of functions $f \in X$ such that $$\max _{g \in X}\left(\min _{0 \leq i \leq 2016}(\max (f(i), g(i)))-\max _{0 \leq i \leq 2016}(\min (f(i), g(i)))\right)=2015$$ | For each $f, g \in X$, we define $$d(f, g):=\min _{0 \leq i \leq 2016}(\max (f(i), g(i)))-\max _{0 \leq i \leq 2016}(\min (f(i), g(i)))$$ Thus we desire $\max _{g \in X} d(f, g)=2015$. First, we count the number of functions $f \in X$ such that $$\exists g: \min _{i} \max \{f(i), g(i)\} \geq 2015 \text { and } \exists ... | 2 \cdot\left(3^{2017}-2^{2017}\right) | HMMT_2 |
[
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 4.5 | A man named Juan has three rectangular solids, each having volume 128. Two of the faces of one solid have areas 4 and 32. Two faces of another solid have areas 64 and 16. Finally, two faces of the last solid have areas 8 and 32. What is the minimum possible exposed surface area of the tallest tower Juan can construct b... | Suppose that $x, y, z$ are the sides of the following solids. Then Volume $=xyz=128$. For the first solid, without loss of generality (with respect to assigning lengths to $x, y, z$), $xy=4$ and $yz=32$. Then $xy^{2}z=128$. Then $y=1$. Solving the remaining equations yields $x=4$ and $z=32$. Then the first solid has di... | 688 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other",
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 4.5 | For each integer $x$ with $1 \leq x \leq 10$, a point is randomly placed at either $(x, 1)$ or $(x,-1)$ with equal probability. What is the expected area of the convex hull of these points? Note: the convex hull of a finite set is the smallest convex polygon containing it. | Let $n=10$. Given a random variable $X$, let $\mathbb{E}(X)$ denote its expected value. If all points are collinear, then the convex hull has area zero. This happens with probability $\frac{2}{2^{n}}$ (either all points are at $y=1$ or all points are at $y=-1$ ). Otherwise, the points form a trapezoid with height 2 (th... | \frac{1793}{128} | HMMT_2 |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 5.25 | For a point $P=(x, y)$ in the Cartesian plane, let $f(P)=\left(x^{2}-y^{2}, 2 x y-y^{2}\right)$. If $S$ is the set of all $P$ so that the sequence $P, f(P), f(f(P)), f(f(f(P))), \ldots$ approaches $(0,0)$, then the area of $S$ can be expressed as $\pi \sqrt{r}$ for some positive real number $r$. Compute $\lfloor 100 r\... | For a point $P=(x, y)$, let $z(P)=x+y \omega$, where $\omega$ is a nontrivial third root of unity. Then $$\begin{aligned} z(f(P))=\left(x^{2}-y^{2}\right)+\left(2 x y-y^{2}\right) \omega=x^{2}+2 x y \omega+y^{2} & (-1-\omega) \\ & =x^{2}+2 x y \omega+y^{2} \omega^{2}=(x+y \omega)^{2}=z(P)^{2} \end{aligned}$$ Applying t... | 133 | HMMT_2 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Algebra -> Linear Algebra -> Matrices"
] | 5.5 | For any positive integer $n, S_{n}$ be the set of all permutations of \{1,2,3, \ldots, n\}. For each permutation $\pi \in S_{n}$, let $f(\pi)$ be the number of ordered pairs $(j, k)$ for which $\pi(j)>\pi(k)$ and $1 \leq j<k \leq n$. Further define $g(\pi)$ to be the number of positive integers $k \leq n$ such that $\p... | Define an $n \times n$ matrix $A_{n}(x)$ with entries $a_{i, j}=x$ if $i \equiv j \pm 1(\bmod n)$ and 1 otherwise. Let $F(x)=\sum_{\pi \in S_{n}}(-1)^{f(\pi)} x^{g(\pi)}$ (here $(-1)^{f(\pi)}$ gives the $\operatorname{sign} \prod \frac{\pi(u)-\pi(v)}{u-v}$ of the permutation $\pi$). Note by construction that $F(x)=\ope... | 995 \times 2^{998} | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 5.5 | A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute the smallest possible value of $n$. | The main claim is that if the light pulse reflects vertically (on the left/right edges) $a$ times and horizontally $b$ times, then $\operatorname{gcd}(a+1, b+1)=1$, and the number of regions is $\frac{(a+2)(b+2)}{2}$. This claim can be conjectured by looking at small values of $a$ and $b$; we give a full proof at the e... | 129 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 5 | Diana is playing a card game against a computer. She starts with a deck consisting of a single card labeled 0.9. Each turn, Diana draws a random card from her deck, while the computer generates a card with a random real number drawn uniformly from the interval $[0,1]$. If the number on Diana's card is larger, she keeps... | By linearity of expectation, we can treat the number of turns each card contributes to the total independently. Let $f(x)$ be the expected number of turns a card of value $x$ contributes (we want $f(0.9)$). If we have a card of value $x$, we lose it after 1 turn with probability $1-x$. If we don't lose it after the fir... | 100 | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 5 | In the Cartesian plane, let $A=(0,0), B=(200,100)$, and $C=(30,330)$. Compute the number of ordered pairs $(x, y)$ of integers so that $\left(x+\frac{1}{2}, y+\frac{1}{2}\right)$ is in the interior of triangle $A B C$. | We use Pick's Theorem, which states that in a lattice polygon with $I$ lattice points in its interior and $B$ lattice points on its boundary, the area is $I+B / 2-1$. Also, call a point center if it is of the form $\left(x+\frac{1}{2}, y+\frac{1}{2}\right)$ for integers $x$ and $y$. The key observation is the following... | 31480 | HMMT_2 |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 4.5 | Let $x_{1}=y_{1}=x_{2}=y_{2}=1$, then for $n \geq 3$ let $x_{n}=x_{n-1} y_{n-2}+x_{n-2} y_{n-1}$ and $y_{n}=y_{n-1} y_{n-2}- x_{n-1} x_{n-2}$. What are the last two digits of $\left|x_{2012}\right|$ ? | Let $z_{n}=y_{n}+x_{n} i$. Then the recursion implies that: $$\begin{aligned} & z_{1}=z_{2}=1+i \\ & z_{n}=z_{n-1} z_{n-2} \end{aligned}$$ This implies that $$z_{n}=\left(z_{1}\right)^{F_{n}}$$ where $F_{n}$ is the $n^{\text {th }}$ Fibonacci number $\left(F_{1}=F_{2}=1\right)$. So, $z_{2012}=(1+i)^{F_{2012}}$. Notice ... | 84 | HMMT_2 |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 5 | The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes. | Let $x=2000$, so the numerator is $$x^{5}+x^{4}+1=\left(x^{2}+x+1\right)\left(x^{3}-x+1\right)$$ (This latter factorization can be noted by the fact that plugging in $\omega$ or $\omega^{2}$ into $x^{5}+x^{4}+1$ gives 0 .) Then $x^{2}+x+1=4002001$ divides the numerator. However, it can easily by checked that 101 doesn'... | 4002001 | HMMT_2 |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5.25 | Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_{1}, x_{2}, \ldots, x_{n}$ for which - $x_{1}^{k}+x_{2}^{k}+\cdots+x_{n}^{k}=1$ for $k=1,2, \ldots, n-1$ - $x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}=2$; and - $x_{1}^{m}+x_{2}^{m}+\cdots+x_{n}^{m}=4$. Compute the smallest possible value of $... | Let $S_{k}=\sum_{j=1}^{n} x_{j}^{k}$, so $S_{1}=S_{2}=\cdots=S_{n-1}=1, S_{n}=2$, and $S_{m}=4$. The first of these conditions gives that $x_{1}, \ldots, x_{n}$ are the roots of $P(x)=x^{n}-x^{n-1}-c$ for some constant $c$. Then $x_{i}^{n}=x_{i}^{n-1}+c$, and thus $$2=S_{n}=S_{n-1}+c n=1+c n$$ so $c=\frac{1}{n}$. Thus,... | 34 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other",
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 7 | Kelvin and 15 other frogs are in a meeting, for a total of 16 frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is cool if for each of the 16 frogs, the number of friends they made during the meeting is a multiple of 4. S... | Consider the multivariate polynomial $$\prod_{1 \leq i<j \leq 16}\left(1+x_{i} x_{j}\right)$$ We're going to filter this by summing over all $4^{16} 16$-tuples $\left(x_{1}, x_{2}, \ldots, x_{16}\right)$ such that $x_{j}= \pm 1, \pm i$. Most of these evaluate to 0 because $i^{2}=(-i)^{2}=-1$, and $1 \cdot-1=-1$. If you... | 1167 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 5 | Five people take a true-or-false test with five questions. Each person randomly guesses on every question. Given that, for each question, a majority of test-takers answered it correctly, let $p$ be the probability that every person answers exactly three questions correctly. Suppose that $p=\frac{a}{2^{b}}$ where $a$ is... | There are a total of $16^{5}$ ways for the people to collectively ace the test. Consider groups of people who share the same problems that they got incorrect. We either have a group of 2 and a group of 3 , or a group 5 . In the first case, we can pick the group of two in $\binom{5}{2}$ ways, the problems they got wrong... | 25517 | HMMT_2 |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 5.5 | Find the number of integers $n$ such that $$ 1+\left\lfloor\frac{100 n}{101}\right\rfloor=\left\lceil\frac{99 n}{100}\right\rceil $$ | Consider $f(n)=\left\lceil\frac{99 n}{100}\right\rceil-\left\lfloor\frac{100 n}{101}\right\rfloor$. Note that $f(n+10100)=\left\lceil\frac{99 n}{100}+99 \cdot 101\right\rceil-\left\lfloor\frac{100 n}{101}+100^{2}\right\rfloor=f(n)+99 \cdot 101-100^{2}=f(n)-1$. Thus, for each residue class $r$ modulo 10100, there is exa... | 10100 | HMMT_2 |
[
"Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Multi-variable",
"Mathematics -> Algebra -> Differential Equations -> Ordinary Differential Equations (ODEs)"
] | 8 | For a continuous and absolutely integrable complex-valued function $f(x)$ on $\mathbb{R}$, define a function $(S f)(x)$ on $\mathbb{R}$ by $(S f)(x)=\int_{-\infty}^{+\infty} e^{2 \pi \mathrm{i} u x} f(u) \mathrm{d} u$. Find explicit forms of $S\left(\frac{1}{1+x^{2}}\right)$ and $S\left(\frac{1}{\left(1+x^{2}\right)^{2... | Write $f(x)=\left(1+x^{2}\right)^{-1}$. For $x \geq 0$, we have $(S f)(x)=\lim _{A \rightarrow+\infty} \int_{-A}^{A} \frac{e^{2 \pi \mathrm{i} u x}}{1+u^{2}} \mathrm{~d} u$. Put $C_{A}:=\{z=u+\mathbf{i} v:-A \leq u \leq A, v=0\} \bigcup\left\{z=A e^{\mathbf{i} \theta}: 0 \leq \theta \leq \pi\right\}$. Note that, $\math... | S\left(\frac{1}{1+x^{2}}\right)=\pi e^{-2 \pi|x|}, S\left(\frac{1}{\left(1+x^{2}\right)^{2}}\right)=\frac{\pi}{2}(1+2 \pi|x|) e^{-2 \pi|x|} | alibaba_global_contest |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 8 | Some squares of a $n \times n$ table $(n>2)$ are black, the rest are white. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers. | The answer is $3n^{2}-5n+2$. The sum attains this value when all squares in even rows are black and the rest are white. It remains to prove that this is the maximum value. The sum in question is the number of pairs of differently coloured squares sharing at least one vertex. There are two kinds of such pairs: sharing a... | 3n^{2}-5n+2 | izho |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Number Theory -> Congruences"
] | 7 | Each of the numbers $1,2, \ldots, 9$ is to be written into one of these circles, so that each circle contains exactly one of these numbers and (i) the sums of the four numbers on each side of the triangle are equal; (ii) the sums of squares of the four numbers on each side of the triangle are equal. Find all ways in wh... | Let $a, b$, and $c$ be the numbers in the vertices of the triangular arrangement. Let $s$ be the sum of the numbers on each side and $t$ be the sum of the squares of the numbers on each side. Summing the numbers (or their squares) on the three sides repeats each once the numbers on the vertices (or their squares): $$\b... | 48 solutions by permuting vertices, adjusting sides, and exchanging middle numbers. | apmoapmo_sol |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of th... | Label the jars $1,2, \ldots, 2017$. I claim that the answer is 2014. To show this, we need both a construction and an upper bound. For the construction, for $1 \leq i \leq 201$, put a coin in the jars $10 i+1,10 i+2, \ldots, 10 i+10$. After this, each of the jars $1,2, \ldots, 2010$ has exactly one coin. Now, put a coi... | 2014 | HMMT_2 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations",
"Mathematics -> Number Theory -> Other"
] | 5 | Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, \ldots, 8,9,10\}$ that satisfy $|S|+\min (S)$. $\max (S)=0$. | Since $\min (S) \cdot \max (S)<0$, we must have $\min (S)=-a$ and $\max (S)=b$ for some positive integers $a$ and $b$. Given $a$ and $b$, there are $|S|-2=a b-2$ elements left to choose, which must come from the set $\{-a+1,-a+2, \ldots, b-2, b-1\}$, which has size $a+b-1$. Therefore the number of possibilities for a g... | 335 | HMMT_2 |
[
"Mathematics -> Geometry -> Plane Geometry -> Angles",
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 5.25 | Let $\omega$ be a fixed circle with radius 1, and let $B C$ be a fixed chord of $\omega$ such that $B C=1$. The locus of the incenter of $A B C$ as $A$ varies along the circumference of $\omega$ bounds a region $\mathcal{R}$ in the plane. Find the area of $\mathcal{R}$. | We will make use of the following lemmas. Lemma 1: If $A B C$ is a triangle with incenter $I$, then $\angle B I C=90+\frac{A}{2}$. Proof: Consider triangle $B I C$. Since $I$ is the intersection of the angle bisectors, $\angle I B C=\frac{B}{2}$ and $\angle I C B=\frac{C}{2}$. It follows that $\angle B I C=180-\frac{B}... | \pi\left(\frac{3-\sqrt{3}}{3}\right)-1 | HMMT_11 |
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 5 | How many ways are there to place four points in the plane such that the set of pairwise distances between the points consists of exactly 2 elements? (Two configurations are the same if one can be obtained from the other via rotation and scaling.) | Let $A, B, C, D$ be the four points. There are 6 pairwise distances, so at least three of them must be equal. Case 1: There is no equilateral triangle. Then WLOG we have $A B=B C=C D=1$. - Subcase 1.1: $A D=1$ as well. Then $A C=B D \neq 1$, so $A B C D$ is a square. - Subcase 1.2: $A D \neq 1$. Then $A C=B D=A D$, so ... | 6 | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4.5 | Consider a permutation $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ of $\{1,2,3,4,5\}$. We say the tuple $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ is flawless if for all $1 \leq i<j<k \leq 5$, the sequence $\left(a_{i}, a_{j}, a_{k}\right)$ is not an arithmetic progression (in that order). Find the number of... | We do casework on the position of 3. - If $a_{1}=3$, then the condition is that 4 must appear after 5 and 2 must appear after 1. It is easy to check there are six ways to do this. - If $a_{2}=3$, then there are no solutions; since there must be an index $i \geq 3$ with $a_{i}=6-a_{1}$. - If $a_{3}=3$, then 3 we must ha... | 20 | HMMT_11 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 7 | Consider $n$ disks $C_{1}, C_{2}, \ldots, C_{n}$ in a plane such that for each $1 \leq i<n$, the center of $C_{i}$ is on the circumference of $C_{i+1}$, and the center of $C_{n}$ is on the circumference of $C_{1}$. Define the score of such an arrangement of $n$ disks to be the number of pairs $(i, j)$ for which $C_{i}$... | The answer is $(n-1)(n-2) / 2$. Let's call a set of $n$ disks satisfying the given conditions an $n$-configuration. For an $n$ configuration $\mathcal{C}=\left\{C_{1}, \ldots, C_{n}\right\}$, let $S_{\mathcal{C}}=\left\{(i, j) \mid C_{i}\right.$ properly contains $\left.C_{j}\right\}$. So, the score of an $n$-configura... | (n-1)(n-2)/2 | apmoapmo_sol |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5.25 | Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color. | Number the vertices 1 through 7 in order. Then, the only way to have three vertices of a regular heptagon that do not form an isosceles triangle is if they are vertices $1,2,4$, rotated or reflected. Thus, it is impossible for have four vertices in the heptagon of one color because it is impossible for all subsets of t... | 294 | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Graph Theory",
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 4.5 | A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies: $f(0)=0$ and $$\left|f\left((n+1) 2^{k}\right)-f\left(n 2^{k}\right)\right| \leq 1$$ for all integers $k \geq 0$ and $n$. What is the maximum possible value of $f(2019)$? | Consider a graph on $\mathbb{Z}$ with an edge between $(n+1) 2^{k}$ and $n 2^{k}$ for all integers $k \geq 0$ and $n$. Each vertex $m$ is given the value $f(m)$. The inequality $\left|f\left((n+1) 2^{k}\right)-f\left(n 2^{k}\right)\right| \leq 1$ means that any two adjacent vertices of this graph must have values which... | 4 | HMMT_11 |
[
"Mathematics -> Applied Mathematics -> Math Word Problems",
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 4.5 | Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain? | Note that after applying the squaring operation, Alice's number will be a perfect square, so she can maximize her score by having a large number of adding operations at the end. However, her scores needs to be large enough that the many additions do not bring her close to a larger square. Hence the strategy is as follo... | 94 | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4.5 | Dorothea has a $3 \times 4$ grid of dots. She colors each dot red, blue, or dark gray. Compute the number of ways Dorothea can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color. | To find an appropriate estimate, we will lower bound the number of rectangles. Let $P(R)$ be the probability a random 3 by 4 grid will have a rectangle with all the same color in the grid. Let $P(r)$ be the probability that a specific rectangle in the grid will have the same color. Note $P(r)=\frac{3}{3^{4}}=\frac{1}{2... | 284688 | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 4.5 | David and Evan are playing a game. Evan thinks of a positive integer $N$ between 1 and 59, inclusive, and David tries to guess it. Each time David makes a guess, Evan will tell him whether the guess is greater than, equal to, or less than $N$. David wants to devise a strategy that will guarantee that he knows $N$ in fi... | We can represent each strategy as a binary tree labeled with the integers from 1 to 59, where David starts at the root and moves to the right child if he is too low and to the left child if he is too high. Our tree must have at most 6 layers as David must guess at most 5 times. Once David has been told that he guessed ... | 36440 | HMMT_11 |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Number Theory -> Prime Numbers"
] | 4.5 | What is the 3-digit number formed by the $9998^{\text {th }}$ through $10000^{\text {th }}$ digits after the decimal point in the decimal expansion of \frac{1}{998}$ ? | Note that \frac{1}{998}+\frac{1}{2}=\frac{250}{499}$ repeats every 498 digits because 499 is prime, so \frac{1}{998}$ does as well (after the first 498 block). Now we need to find $38^{\text {th }}$ to $40^{\text {th }}$ digits. We expand this as a geometric series $$\frac{1}{998}=\frac{\frac{1}{1000}}{1-\frac{2}{1000}... | 042 | HMMT_11 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 4.5 | There are six empty slots corresponding to the digits of a six-digit number. Claire and William take turns rolling a standard six-sided die, with Claire going first. They alternate with each roll until they have each rolled three times. After a player rolls, they place the number from their die roll into a remaining em... | A number being divisible by 6 is equivalent to the following two conditions: - the sum of the digits is divisible by 3 - the last digit is even Regardless of Claire and William's strategies, the first condition is satisfied with probability $\frac{1}{3}$. So Claire simply plays to maximize the chance of the last digit ... | \frac{43}{192} | HMMT_11 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 4.5 | An equiangular hexagon has side lengths $1,1, a, 1,1, a$ in that order. Given that there exists a circle that intersects the hexagon at 12 distinct points, we have $M<a<N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$. | We claim that the greatest possible value of $M$ is $\sqrt{3}-1$, whereas the least possible value of $N$ is 3 . To begin, note that the condition requires the circle to intersect each side of the hexagon at two points on its interior. This implies that the center must be inside the hexagon as its projection onto all s... | \frac{3 \sqrt{3}+3}{2} | HMMT_11 |
[
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 7 | Let $r_{k}$ denote the remainder when $\binom{127}{k}$ is divided by 8. Compute $r_{1}+2 r_{2}+3 r_{3}+\cdots+63 r_{63}$. | Let $p_{k}=\frac{128-k}{k}$, so $$\binom{127}{k}=p_{1} p_{2} \cdots p_{k}$$ Now, for $k \leq 63$, unless $32 \mid \operatorname{gcd}(k, 128-k)=\operatorname{gcd}(k, 128), p_{k} \equiv-1(\bmod 8)$. We have $p_{32}=\frac{96}{32}=3$. Thus, we have the following characterization: $$r_{k}= \begin{cases}1 & \text { if } k \t... | 8096 | HMMT_11 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 5 | Let $A B C D E F$ be a convex hexagon with the following properties. (a) $\overline{A C}$ and $\overline{A E}$ trisect $\angle B A F$. (b) $\overline{B E} \| \overline{C D}$ and $\overline{C F} \| \overline{D E}$. (c) $A B=2 A C=4 A E=8 A F$. Suppose that quadrilaterals $A C D E$ and $A D E F$ have area 2014 and 1400, ... | From conditions (a) and (c), we know that triangles $A F E, A E C$ and $A C B$ are similar to one another, each being twice as large as the preceding one in each dimension. Let $\overline{A E} \cap \overline{F C}=P$ and $\overline{A C} \cap \overline{E B}=Q$. Then, since the quadrilaterals $A F E C$ and $A E C B$ are s... | 7295 | HMMT_11 |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 5.25 | In acute $\triangle A B C$ with centroid $G, A B=22$ and $A C=19$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to $A C$ and $A B$ respectively. Let $G^{\prime}$ be the reflection of $G$ over $B C$. If $E, F, G$, and $G^{\prime}$ lie on a circle, compute $B C$. | Note that $B, C, E, F$ lie on a circle. Moreover, since $B C$ bisects $G G^{\prime}$, the center of the circle that goes through $E, F, G, G^{\prime}$ must lie on $B C$. Therefore, $B, C, E, F, G, G^{\prime}$ lie on a circle. Specifically, the center of this circle is $M$, the midpoint of $B C$, as $M E=M F$ because $M... | 13 | HMMT_11 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5.25 | Each square in a $3 \times 10$ grid is colored black or white. Let $N$ be the number of ways this can be done in such a way that no five squares in an 'X' configuration (as shown by the black squares below) are all white or all black. Determine $\sqrt{N}$. | Note that we may label half of the cells in our board the number 0 and the other half 1, in such a way that squares labeled 0 are adjacent only to squares labeled 1 and vice versa. In other words, we make this labeling in a 'checkerboard' pattern. Since cells in an 'X' formation are all labeled with the same number, th... | 25636 | HMMT_11 |
[
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 4 | Consider a $10 \times 10$ grid of squares. One day, Daniel drops a burrito in the top left square, where a wingless pigeon happens to be looking for food. Every minute, if the pigeon and the burrito are in the same square, the pigeon will eat $10 \%$ of the burrito's original size and accidentally throw it into a rando... | Label the squares using coordinates, letting the top left corner be $(0,0)$. The burrito will end up in 10 (not necessarily different) squares. Call them $p_{1}=\left(x_{1}, y_{1}\right)=(0,0), p_{2}=\left(x_{2}, y_{2}\right), \ldots, p_{10}=\left(x_{10}, y_{10}\right)$. $p_{2}$ through $p_{10}$ are uniformly distribut... | 71.8 | HMMT_11 |
[
"Mathematics -> Number Theory -> Base Representations -> Other"
] | 4.5 | Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \neq 4$ such that the base -4 representation of $n$ is the same as the base $b$ representation of $n$. | All 1 digit numbers, $0,1,2,3$, are solutions when, say, $b=5$. (Of course, $d \in \{0,1,2,3\}$ works for any base $b$ of absolute value greater than $d$ but not equal to 4 .) Consider now positive integers $n=\left(a_{d} \ldots a_{1} a_{0}\right)_{4}$ with more than one digit, so $d \geq 1, a_{d} \neq 0$, and $0 \leq ... | 1026 | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4 | Let $S$ be a subset with four elements chosen from \{1,2, \ldots, 10\}$. Michael notes that there is a way to label the vertices of a square with elements from $S$ such that no two vertices have the same label, and the labels adjacent to any side of the square differ by at least 4 . How many possibilities are there for... | Let the four numbers be $a, b, c, d$ around the square. Assume without loss of generality that $a$ is the largest number, so that $a>b$ and $a>d$. Note that $c$ cannot be simultaneously smaller than one of $b, d$ and larger than the other because, e.g. if $b>c>d$, then $a>b>c>d$ and $a \geq d+12$. Hence $c$ is either s... | 36 | HMMT_11 |
[
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 4.5 | A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheres of radius $x$ are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of $x$? | Let $O$ be the center of the large sphere, and let $O_{1}, O_{2}, O_{3}$ be the centers of the small spheres. Consider $G$, the center of equilateral $\triangle O_{1} O_{2} O_{3}$. Then if the radii of the small spheres are $r$, we have that $O G=8+r$ and $O_{1} O_{2}=O_{2} O_{3}=O_{3} O_{1}=2 r$, implying that $O_{1} ... | \frac{15 \sqrt{37}-75}{4} | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 4 | In Middle-Earth, nine cities form a 3 by 3 grid. The top left city is the capital of Gondor and the bottom right city is the capital of Mordor. How many ways can the remaining cities be divided among the two nations such that all cities in a country can be reached from its capital via the grid-lines without passing thr... | For convenience, we will center the grid on the origin of the coordinate plane and align the outer corners of the grid with the points $( \pm 1, \pm 1)$, so that $(-1,1)$ is the capital of Gondor and $(1,-1)$ is the capital of Mordor. We will use casework on which nation the city at $(0,0)$ is part of. Assume that is b... | 30 | HMMT_11 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 4 | 20 players are playing in a Super Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays 18 distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players tha... | Consider instead the complement of the tournament: The 10 possible matches that are not played. In order for each player to play 18 games in the tournament, each must appear once in these 10 unplayed matches. Players $n$ and $n+1$ will win the same number of games if, in the matching, they are matched with each other, ... | 4 | HMMT_11 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 5 | The taxicab distance between points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is $\left|x_{2}-x_{1}\right|+\left|y_{2}-y_{1}\right|$. A regular octagon is positioned in the $x y$ plane so that one of its sides has endpoints $(0,0)$ and $(1,0)$. Let $S$ be the set of all points inside the octagon whose... | In the taxicab metric, the set of points that lie at most $d$ units away from some fixed point $P$ form a square centered at $P$ with vertices at a distance of $d$ from $P$ in directions parallel to the axes. The diagram above depicts the intersection of an octagon with eight such squares for $d=\frac{2}{3}$ centered a... | 2309 | HMMT_11 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4 | There are 21 competitors with distinct skill levels numbered $1,2, \ldots, 21$. They participate in a pingpong tournament as follows. First, a random competitor is chosen to be "active", while the rest are "inactive." Every round, a random inactive competitor is chosen to play against the current active one. The player... | Solution 1: Insert a player with skill level 0, who will be the first active player (and lose their first game). If Alice plays after any of the players with skill level $12,13, \ldots, 21$, which happens with probability $\frac{10}{11}$, then she will play exactly 1 game. If Alice is the first of the players with skil... | \frac{47}{42} | HMMT_11 |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 4 | Alice and Bob are playing in the forest. They have six sticks of length $1,2,3,4,5,6$ inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of this hexagon. | Let the side lengths, in counterclockwise order, be $a, b, c, d, e, f$. Place the hexagon on the coordinate plane with edge $a$ parallel to the $x$-axis and the intersection between edge $a$ and edge $f$ at the origin (oriented so that edge $b$ lies in the first quadrant). If you travel along all six sides of the hexag... | 33 \sqrt{3} | HMMT_11 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other",
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 5 | 3000 people each go into one of three rooms randomly. What is the most likely value for the maximum number of people in any of the rooms? Your score for this problem will be 0 if you write down a number less than or equal to 1000. Otherwise, it will be $25-27 \frac{|A-C|}{\min (A, C)-1000}$. | To get a rough approximation, we can use the fact that a sum of identical random variables converges to a Gaussian distribution in this case with a mean of 1000 and a variance of $3000 \cdot \frac{2}{9}=667$. Since $\sqrt{667} \approx 26,1026$ is a good guess, as Gaussians tend to differ from their mean by approximatel... | 1019 | HMMT_2 |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Number Theory -> Factorization"
] | 6 | Call a positive integer $n$ quixotic if the value of $\operatorname{lcm}(1,2,3, \ldots, n) \cdot\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\right)$ is divisible by 45 . Compute the tenth smallest quixotic integer. | Let $L=\operatorname{lcm}(1,2,3, \ldots, n)$, and let $E=L\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)$ denote the expression. In order for $n$ to be quixotic, we need $E \equiv 0(\bmod 5)$ and $E \equiv 0(\bmod 9)$. We consider these two conditions separately. Claim: $E \equiv 0(\bmod 5)$ if and only if $... | 573 | HMMT_11 |
[
"Mathematics -> Algebra -> Number Theory -> Other",
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 4.5 | For positive integers $n$, let $f(n)$ be the product of the digits of $n$. Find the largest positive integer $m$ such that $$\sum_{n=1}^{\infty} \frac{f(n)}{m\left\lfloor\log _{10} n\right\rfloor}$$ is an integer. | We know that if $S_{\ell}$ is the set of all positive integers with $\ell$ digits, then $$\begin{aligned} & \sum_{n \in S_{\ell}} \frac{f(n)}{k^{\left\lfloor\log _{10}(n)\right\rfloor}}=\sum_{n \in S_{\ell}} \frac{f(n)}{k^{\ell-1}}=\frac{(0+1+2+\ldots+9)^{\ell}}{k^{\ell-1}}= \\ & 45 \cdot\left(\frac{45}{k}\right)^{\ell... | 2070 | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 5 | On a chessboard, a queen attacks every square it can reach by moving from its current square along a row, column, or diagonal without passing through a different square that is occupied by a chess piece. Find the number of ways in which three indistinguishable queens can be placed on an $8 \times 8$ chess board so that... | The configuration of three cells must come in a 45-45-90 triangle. There are two cases, both shown above: the triangle has legs parallel to the axes, or it has its hypotenuse parallel to an axis. The first case can be solved by noticing that each selection of four cells in the shape of a square corresponds to four such... | 864 | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 4 | Eight points are chosen on the circumference of a circle, labelled $P_{1}, P_{2}, \ldots, P_{8}$ in clockwise order. A route is a sequence of at least two points $P_{a_{1}}, P_{a_{2}}, \ldots, P_{a_{n}}$ such that if an ant were to visit these points in their given order, starting at $P_{a_{1}}$ and ending at $P_{a_{n}... | Solution 1: How many routes are there if we are restricted to $n$ available points, and we must use all $n$ of them? The answer is $n 2^{n-2}$ : first choose the starting point, then each move after that must visit one of the two neighbors of your expanding region of visited points (doing anything else would prevent yo... | 8744 | HMMT_11 |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | The skeletal structure of coronene, a hydrocarbon with the chemical formula $\mathrm{C}_{24} \mathrm{H}_{12}$, is shown below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connected to it and each hydrogen $(\mathrm{H})$ requires exactly one b... | Note that each carbon needs exactly one double bond. Label the six carbons in the center $1,2,3,4,5,6$ clockwise. We consider how these six carbons are double-bonded. If a carbon in the center is not double-bonded to another carbon in the center, it must double-bond to the corresponding carbon on the outer ring. This w... | 20 | HMMT_11 |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 5 | A string of digits is defined to be similar to another string of digits if it can be obtained by reversing some contiguous substring of the original string. For example, the strings 101 and 110 are similar, but the strings 3443 and 4334 are not. (Note that a string is always similar to itself.) Consider the string of d... | We first count the number of substrings that one could pick to reverse to yield a new substring. If we insert two dividers into the sequence of 50 digits, each arrangement of 2 dividers among the 52 total objects specifies a substring that is contained between the two dividers, for a total of $\binom{52}{2}$ substrings... | 1126 | HMMT_11 |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 5 | Compute the number of positive integers less than 10! which can be expressed as the sum of at most 4 (not necessarily distinct) factorials. | Since $0!=1!=1$, we ignore any possible 0!'s in our sums. Call a sum of factorials reduced if for all positive integers $k$, the term $k$! appears at most $k$ times. It is straightforward to show that every positive integer can be written uniquely as a reduced sum of factorials. Moreover, by repeatedly replacing $k+1$ ... | 648 | HMMT_11 |
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