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|---|---|---|---|
Given that \( x \geq 1 \), \( y \geq 1 \), and \(\lg ^{2} x+\lg ^{2} y=\lg 10 x^{2}+\lg 10 y^{2}\), find the maximum value of \( u=\lg x y \).
|
2 + 2 \sqrt{2}
|
olympiads
| 0.0625 |
In which cases is the congruence $a x \equiv b \pmod{m}$ solvable? Describe all solutions to this congruence in integers.
|
x = x_0 + k \left( \frac{m}{d} \right) \text{ for } k \in \mathbb{Z}
|
olympiads
| 0.34375 |
After a football match, the coach lined up the team as shown in the picture and commanded: "Those with a number smaller than any of their neighbors run to the locker room." After several people ran away, he repeated his command. The coach continued until only one player remained. What is Igor's number if it is known that after he ran away, 3 people remained in the lineup? (After each command, one or more players would run away, and the lineup would close up, leaving no empty spots between the remaining players.)
|
5
|
olympiads
| 0.203125 |
Move 1 matchstick to make the equation true. Write the modified equation on the line:
(Note: In the equation, " $\mathrm{Z}$ " stands for 2)
|
3 = 11 + 1 - 2 - 7
|
olympiads
| 0.0625 |
Find all values of the real parameter \( p \) for which the following system of equations is solvable:
\[
\left\{\begin{aligned}
2[x] + y & = \frac{3}{2} \\
3[x] - 2y & = p
\end{aligned}\right.
\]
Here, \([a]\) denotes the integer part of the number \( a \).
|
p = 7k - 3, \quad k \in \mathbb{Z}
|
olympiads
| 0.34375 |
Let \( a \) and \( b \) be integer solutions to \( 17a + 6b = 13 \). What is the smallest possible positive value for \( a - b \)?
|
17
|
olympiads
| 0.09375 |
Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds: $$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$
where $L_{\max}$ , $L_{\min}$ is the maximal and minimal distance between chosen points.
|
4
|
aops_forum
| 0.125 |
Represent the number 231 as a sum of several natural numbers such that the product of these terms also equals 231.
|
231 = 3 + 7 + 11 + \underbrace{1 + 1 + \cdots + 1}_{210 \text{ ones}}
|
olympiads
| 0.21875 |
Find a two-digit number equal to the sum of the cube of its tens digit and the square of its units digit.
|
24
|
olympiads
| 0.109375 |
Aunt Masha is three years younger than the combined age of Sasha and his peer Pasha. How old is Sasha now if Aunt Masha was once as old as Pasha is now?
|
3
|
olympiads
| 0.4375 |
We folded a rectangular sheet of paper along one of its diagonals. After the fold, the four corners became vertices of a trapezoid with three equal sides. What is the length of the shorter side of the original rectangle if the longer side is 12 cm?
|
4\sqrt{3} ext{ cm}
|
olympiads
| 0.09375 |
Each of the five vases shown has the same height and each has a volume of 1 liter. Half a liter of water is poured into each vase. In which vase would the level of the water be the highest?
|
A
|
olympiads
| 0.0625 |
Let $G$ be the centroid of triangle $ABC$ with $AB=13,BC=14,CA=15$ . Calculate the sum of the distances from $G$ to the three sides of the triangle.
Note: The *centroid* of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side.
|
\frac{2348}{195}
|
aops_forum
| 0.203125 |
Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length \(\frac{1}{2}\) and edges parallel to those of the unit cube that contains all three points?
|
\frac{1}{8}
|
olympiads
| 0.09375 |
Find all infinite bounded sequences of natural numbers \( a_1, a_2, a_3, \ldots \), in which, for all terms starting from the third one, the following condition is satisfied:
\[
a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}
\]
|
a_1 = a_2 = \ldots = 2
|
olympiads
| 0.09375 |
Calculate the surface integral
$$
\iint_{\Sigma}\left(x^{2}+y^{2}\right) d \sigma
$$
where $\Sigma$ is the part of the surface $x^{2}+y^{2}=1$ bounded by the planes $z=0$ and $z=2$.
|
4 \pi
|
olympiads
| 0.34375 |
\[
\left\{\begin{array}{l}
\log _{x y} \frac{y}{x}-\log _{y}^{2} x=1, \\
\log _{2}(y-x)=1
\end{array}\right.
\]
|
(1, 3)
|
olympiads
| 0.1875 |
Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$ .
|
6
|
aops_forum
| 0.0625 |
\(\sin 2A + \sin 2B + \sin 2C = 4 \sin A \sin B \sin C\).
|
4 \sin A \sin B \sin C
|
olympiads
| 0.078125 |
Which quadratic equation in the form \( x^2 + ax + b = 0 \) has roots that are equal to its coefficients?
|
Option 3: x^2 + x - 2 = 0, \, x_1 = 1, \, x_2 = -2
|
olympiads
| 0.0625 |
All angles of $ABC$ are in $(30,90)$ . Circumcenter of $ABC$ is $O$ and circumradius is $R$ . Point $K$ is projection of $O$ to angle bisector of $\angle B$ , point $M$ is midpoint $AC$ . It is known, that $2KM=R$ . Find $\angle B$
|
60^ extcirc}
|
aops_forum
| 0.171875 |
How many integers from 1 to 300 can be divided by 7 and also be divisible by either 2 or 5?
|
25
|
olympiads
| 0.109375 |
In a number matrix as shown, the three numbers in each row are in arithmetic progression, and the three numbers in each column are also in arithmetic progression. Given that \( a_{22} = 2 \), find the sum of all 9 numbers in the matrix.
|
18
|
olympiads
| 0.4375 |
The work ratios of projects A, B, and C are $1: 2: 3$. They are undertaken by three teams, Team 1, Team 2, and Team 3, respectively, all starting at the same time. After a certain number of days, the amount of work completed by Team 1 is half of the amount of work not completed by Team 2, the amount of work completed by Team 2 is one-third of the amount of work not completed by Team 3, and the amount of work completed by Team 3 equals the amount of work not completed by Team 1. What is the ratio of the work efficiencies of Teams 1, 2, and 3?
|
\frac{1}{a}:\frac{1}{b}:\frac{1}{c} = 4:6:3
|
olympiads
| 0.09375 |
5 printing machines can print 5 books in 5 days. If \( n \) printing machines are required in order to have 100 books printed in 100 days, find \( n \).
|
5
|
olympiads
| 0.4375 |
Find all pairs of positive integers \((x, y)\) such that \(7^{x} - 3 \cdot 2^{y} = 1\).
|
(1, 1) \text{ and } (2, 4)
|
olympiads
| 0.21875 |
Xiao Ming's father needs to first ride a mountain bike for 10 minutes and then take the subway for 40 minutes to get to work every day. One day, the subway broke down, so he directly rode the bike to work, which took a total of 3.5 hours. What is the ratio of the subway's speed to the mountain bike's speed?
|
5
|
olympiads
| 0.125 |
Tommy takes a 25-question true-false test. He answers each question correctly with independent probability $\frac{1}{2}$ . Tommy earns bonus points for correct streaks: the first question in a streak is worth 1 point, the second question is worth 2 points, and so on. For instance, the sequence TFFTTTFT is worth 1 + 1 + 2 + 3 + 1 = 8 points. Compute the expected value of Tommy’s score.
|
50
|
aops_forum
| 0.0625 |
A triangle with sides \(a, b\), and \(c\) is given. Denote by \(s\) the semiperimeter, that is \(s = \frac{a+b+c}{2}\). Construct a triangle with sides \(s-a, s-b\), and \(s-c\). This process is repeated until a triangle can no longer be constructed with the given side lengths.
For which original triangles can this process be repeated indefinitely?
|
Only equilateral triangles
|
olympiads
| 0.109375 |
Determine the largest positive integer \( n \) with \( n < 500 \) for which \( 6048 \cdot 28^n \) is a perfect cube (that is, it is equal to \( m^3 \) for some positive integer \( m \) ).
|
497
|
olympiads
| 0.171875 |
Given a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ with $a > 1$ and $b > 0$, the focal distance is $2c$. A line $l$ passes through the points $(a, 0)$ and $(0, b)$, and the sum of the distances from the points $(1,0)$ and $(-1,0)$ to the line $l$ is $s \geqslant \frac{4}{5}c$. Determine the range of the eccentricity $e$ of the hyperbola.
|
\left[\frac{\sqrt{5}}{2}, \sqrt{5}\right]
|
olympiads
| 0.09375 |
There are three triangles: an acute triangle, a right triangle, and an obtuse triangle. Sasha took one triangle, and Borya took the remaining two. It turned out that Borya can place one of his triangles against the other (without overlapping) to form a triangle identical to Sasha's. Which triangle did Sasha take?
|
Right
|
olympiads
| 0.15625 |
For the pie fillings, there are rice, meat, and eggs available. How many different fillings can be made from these ingredients? (Remember that the filling can be made from a different number of ingredients).
|
7
|
olympiads
| 0.203125 |
Given the real number \(a\) such that there exists exactly one square with all four vertices lying on the curve \(y = x^3 + ax\). Determine the side length of this square.
|
\sqrt[4]{72}
|
olympiads
| 0.0625 |
Determine the smallest positive integer that has 144 positive divisors, and among these divisors, there are ten consecutive ones.
|
110880
|
olympiads
| 0.09375 |
In the cells of an \( n \times n \) grid, where \( n > 1 \), different integers from 1 to \( n^2 \) must be placed so that every two consecutive numbers are found in adjacent cells (sharing a side), and every two numbers that give the same remainder when divided by \( n \) are in different rows and different columns. For which \( n \) is this possible?
|
All even n
|
olympiads
| 0.078125 |
Given: Three vertices \( A, B, \) and \( C \) of a square are on the parabola \( y = x^2 \).
Find: The minimum possible area of such a square.
|
2
|
olympiads
| 0.09375 |
In a plane rectangular coordinate system, if there are exactly three lines that are at a distance of 1 from point $A(2,2)$ and at a distance of 3 from point $B(m, 0)$, then the set of values of the real number $m$ is $\qquad$.
|
\{2 - 2\sqrt{3}, 2 + 2\sqrt{3}\}
|
olympiads
| 0.09375 |
For which integers \( n \geq 8 \) is \( n + \frac{1}{n-7} \) an integer?
|
8
|
olympiads
| 0.171875 |
For any positive integer $n$, let $d(n)$ represent the number of positive divisors of $n$ (including 1 and $n$ itself). Determine all possible positive integers $K$ such that there exists a positive integer $n$ satisfying $\frac{d\left(n^{2}\right)}{d(n)} = K$.
|
All positive odd integers
|
olympiads
| 0.0625 |
Find all values of the digit \( a \) if the number \(\overline{a719}\) is divisible by 11.
|
4
|
olympiads
| 0.15625 |
Four girls: Anna, Bea, Cili, and Dóra, sang songs at the music school's exam. In every song, three of them sang while the fourth one accompanied them on the piano. Anna sang the most, a total of eight times, and Dóra sang the least, only five times - the others each sang more than this.
How many songs were performed in total?
|
9
|
olympiads
| 0.140625 |
A marksman shoots at a target located $d$ meters away. An observer standing $a$ meters from the marksman and $b$ meters from the target (as shown in Figure 13.10) hears the sound of the shot and the sound of the bullet hitting the target simultaneously. Determine the speed of the bullet, given that the speed of sound is $v$ meters per second.
|
\frac{dv}{a - b}
|
olympiads
| 0.0625 |
The equation \( x^2 + ax + 3 = 0 \) has two distinct roots \( x_1 \) and \( x_2 \). Additionally, it is given that:
\[ x_1^3 - \frac{39}{x_2} = x_2^3 - \frac{39}{x_1} \]
Find all possible values of \( a \).
|
\pm 4
|
olympiads
| 0.125 |
A student "illegally" simplified the exponents but got the correct answer:
\[
\frac{43^{3}+17^{3}}{43^{3}+26^{3}}=\frac{43+17}{43+26}=\frac{20}{23}
\]
Explain the reason for this "coincidence."
|
\frac{20}{23}
|
olympiads
| 0.46875 |
There are 2 coins in a wallet with a total value of 15 kopecks. One of them is not a five-kopeck coin. What are these coins?
|
5\ \text{kopecks}\ \text{and}\ \ 10\ \text{kopecks}
|
olympiads
| 0.4375 |
In the 2002 tax return, for those whose annual gross income was more than 1,050,000 forints, they had to pay 267,000 forints in tax plus 40% of the amount exceeding 1,050,000 forints. What monthly gross income would result in the income tax being 30% of the income?
|
127,500
|
olympiads
| 0.546875 |
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that
\[
f(a-b) f(c-d) + f(a-d) f(b-c) \leq (a-c) f(b-d)
\]
for all real numbers \(a, b, c,\) and \(d\).
|
f(x) = 0 \text{ or } f(x) = x
|
olympiads
| 0.296875 |
Find the coordinates of point $A$ that is equidistant from points $B$ and $C$.
$A(0 ; y ; 0)$
$B(1 ; 6 ; 4)$
$C(5 ; 7 ; 1)$
|
(0, 11, 0)
|
olympiads
| 0.40625 |
In a convex quadrilateral \(ABCD\), \(AB = 10\), \(CD = 15\). The diagonals \(AC\) and \(BD\) intersect at point \(O\), with \(AC = 20\). The triangles \(AOD\) and \(BOC\) have equal areas. Find \(AO\).
|
8
|
olympiads
| 0.078125 |
Let $ f(x)$ be a function such that $ 1\minus{}f(x)\equal{}f(1\minus{}x)$ for $ 0\leq x\leq 1$ .
Evaluate $ \int_0^1 f(x)\ dx$ .
|
\frac{1}{2}
|
aops_forum
| 0.59375 |
In the sequence $\left\{a_{n}\right\}$, if $a_{1} = 3$ and $n a_{n} = (n + 1) a_{n+1}$, then $a_{n} = ?$
|
\frac{3}{n}
|
olympiads
| 0.3125 |
Given that \( 100^{2} + 1^{2} = 65^{2} + 76^{2} = pq \) for some primes \( p \) and \( q \). Find \( p + q \).
|
210
|
olympiads
| 0.28125 |
Find all triples of positive integers $(x, y, z)$ with $$ \frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3 $$
|
(1, 1, 1)
|
aops_forum
| 0.25 |
Construct a triangle given two sides and the bisector of the angle between them.
|
The triangle ABC is successfully constructed based on the given parameters.
|
olympiads
| 0.09375 |
For each vertex of triangle $ABC$ , the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle.
|
\gamma = 120^\circ
|
aops_forum
| 0.21875 |
On average, how many times do you need to roll a fair six-sided die for the sum of the obtained numbers to be at least 3?
|
1.36
|
olympiads
| 0.078125 |
Expand the function \( f(x)=x^{3}-4 x^{2}+5 x-2 \) using the Taylor series formula in powers of \( (x+1) \), i.e., with the expansion centered at the point \( x_{0}=-1 \).
|
f(x) = -12 + 16(x + 1) - 7(x + 1)^2 + (x + 1)^3
|
olympiads
| 0.53125 |
In each cell of an \( n \times n \) table, there is a natural number. It turns out that for all \( k \) from 1 to \( n \), the sum of the numbers in the \( k \)-th column differs by one from the sum of the numbers in the \( k \)-th row. For which \( n \) is this possible?
|
even n
|
olympiads
| 0.0625 |
According to a survey conducted in the 7th "E" grade class, it was found that $20\%$ of the students who are interested in mathematics are also interested in physics, and $25\%$ of the students who are interested in physics are also interested in mathematics. Only Peter and Vasya are not interested in either of these subjects. How many students are there in the 7th "E" grade class, given that there are more than 20 but less than 30 of them?
|
26
|
olympiads
| 0.09375 |
For the function \( y = (n-1) t^2 - 10t + 10, \; t \in (0,4] \), does there exist a positive integer \( n \) such that \( 0 < y \leq 30 \) always holds?
|
4
|
olympiads
| 0.1875 |
Let \( p \) be an arbitrary (positive) prime number. Find all positive integer values of \( b \) for which the quadratic equation \( x^2 - bx + bp = 0 \) has integer roots.
|
(p + 1)^2 \text{ or } 4p
|
olympiads
| 0.15625 |
A tree is 24 meters tall. A snail at the bottom of the tree wants to climb to the top. During the day, it climbs 6 meters up, and at night, it slides down 4 meters. After how many days can the snail reach the top of the tree?
|
10
|
olympiads
| 0.171875 |
Anna, Vera, Katya, and Natasha competed in a long jump competition. When asked about their placements, they responded:
Anna: - I was neither first nor last.
Vera: - I was not last.
Katya: - I was first.
Natasha: - I was last.
It is known that one of the girls lied, and three told the truth. Who took first place?
|
ext{Vera}
|
olympiads
| 0.0625 |
In an isosceles triangle \( A B C \), angle \( B \) is a right angle, and \( A B = B C = 2 \). A circle touches both legs at their midpoints and intersects the hypotenuse at chord \( D E \). Find the area of triangle \( B D E \).
|
\sqrt{2}
|
olympiads
| 0.078125 |
In different historical periods, the conversion between "jin" and "liang" was different. The idiom "ban jin ba liang" comes from the 16-based system. For convenience, we assume that in ancient times, 16 liang equaled 1 jin, with each jin being equivalent to 600 grams in today's terms. Currently, 10 liang equals 1 jin, with each jin equivalent to 500 grams today. There is a batch of medicine, with part weighed using the ancient system and the other part using the current system, and it was found that the sum of the number of "jin" is 5 and the sum of the number of "liang" is 68. How many grams is this batch of medicine in total?
|
2800
|
olympiads
| 0.40625 |
Who is who? In front of us are two inhabitants of the country of knights and liars, $\mathcal{A}$ and $B$. $\mathcal{A}$ says: "I am a liar and $B$ is not a liar." Who among the islanders, $\mathcal{A}$ and $B$, is a knight and who is a liar?
|
\text{$\mathcal{A}$ is a liar and $B$ is a liar.}
|
olympiads
| 0.109375 |
Among 25 students in a group containing 10 girls, 5 tickets are being drawn. Find the probability that 2 of the ticket holders are girls.
|
0.385
|
olympiads
| 0.09375 |
$ABCDEF$ is a regular hexagon with a mirrored inner surface. A light ray originates from point $A$ and, after two reflections off the sides of the hexagon (at points $M$ and $N$), reaches point $D$. Find the tangent of angle $EAM$.
|
\frac{1}{3 \sqrt{3}}
|
olympiads
| 0.234375 |
The volume of the tetrahedron \( A B C D \) is \( V \). Points \( K \), \( L \), and \( M \) are taken on the edges \( C D \), \( D B \), and \( A B \) respectively, such that \( 2 C K = C D \), \( 3 D L = D B \), \( 5 B M = 2 A B \). Find the volume of the tetrahedron \( K L M D \).
|
\frac{V}{15}
|
olympiads
| 0.0625 |
For which values of \(a, b, c\) is the polynomial \(x^{4} + a x^{2} + b x + c\) exactly divisible by \((x-1)^{3}\)?
|
a = -6, b = 8, c = -3
|
olympiads
| 0.484375 |
$N^{3}$ unit cubes are drilled diagonally and tightly strung on a thread, which is then tied into a loop (i.e., the vertex of the first cube is connected to the vertex of the last cube). For which $N$ can such a necklace of cubes be packed into a cubic box with an edge length of $N$?
|
N is even
|
olympiads
| 0.0625 |
What is the minimum number of cells that need to be marked on a 5 by 5 grid so that among the marked cells, there are no neighbors (sharing a common side or a common vertex), and adding any one cell to these cells would violate this condition?
|
4
|
olympiads
| 0.0625 |
Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will be satisfied with the service in this online store.
|
0.64
|
olympiads
| 0.0625 |
In the Cartesian coordinate system $xOy$, given points $M(-1,2)$ and $N(1,4)$, point $P$ moves along the x-axis. When the angle $\angle MPN$ reaches its maximum value, the x-coordinate of point $P$ is ______.
|
1
|
olympiads
| 0.203125 |
The sequence of numbers \( x_{1}, x_{2}, \ldots \) is such that \( x_{1} = \frac{1}{2} \) and \( x_{k+1} = x_{k}^{2} + x_{k} \) for all natural numbers \( k \). Find the integer part of the sum \( \frac{1}{x_{1}+1} + \frac{1}{x_{2}+1} + \ldots + \frac{1}{x_{100}+1} \).
|
1
|
olympiads
| 0.125 |
A rectangular plot for rye cultivation was modified: one side was increased by 20%, while the other side was decreased by 20%. Will the rye yield change as a result, and if so, by how much?
|
Decreases by 4\%
|
olympiads
| 0.3125 |
On a Pacific beach, an electronic shark warning system, which on average triggers an alert once a month (specifically, 1 day out of 30), has been installed. The false alarms occur ten times more frequently than the instances when real shark appearances are not detected by the system. Additionally, it is known that the system detects only three out of four shark appearances. What percentage of days on this beach are "ordinary" days, i.e., days when sharks do not appear and false alarms do not occur?
|
64.44\%
|
olympiads
| 0.0625 |
A cyclist and a motorist simultaneously set off from point $A$ to point $B$, with the motorist traveling 5 times faster than the cyclist. However, halfway through the journey, the car broke down, and the motorist had to continue to $B$ on foot at a speed that is half that of the cyclist. Who arrived at $B$ first?
|
Cyclist
|
olympiads
| 0.390625 |
Find the smallest natural number \( n \) such that when any 5 vertices of a regular \( n \)-gon \( S \) are colored red, there is always an axis of symmetry \( l \) of \( S \) for which the reflection of every red vertex across \( l \) is not a red vertex.
|
14
|
olympiads
| 0.203125 |
Calculate the sum:
\[
\frac{1^{2}+2^{2}}{1 \times 2}+\frac{2^{2}+3^{2}}{2 \times 3}+\ldots+\frac{100^{2}+101^{2}}{100 \times 101}
\]
(where \( n^{2} = n \times n \)).
|
200 \frac{100}{101}
|
olympiads
| 0.40625 |
Xiaoming encountered three people: a priest, a liar, and a lunatic. The priest always tells the truth, the liar always lies, and the lunatic sometimes tells the truth and sometimes lies. The first person said, "I am the lunatic." The second person said, "You are lying, you are not the lunatic!" The third person said, "Stop talking, I am the lunatic." Who among these three people is the lunatic?
|
3
|
olympiads
| 0.1875 |
Let the maximum and minimum elements of the set $\left\{\left.\frac{5}{x}+y \right\rvert\, 1 \leqslant x \leqslant y \leqslant 5\right\}$ be $M$ and $N$, respectively. Find $M \cdot N$.
|
20\sqrt{5}
|
olympiads
| 0.109375 |
For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$ th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$ .
|
40320
|
aops_forum
| 0.125 |
In parallelogram \(ABCD\), diagonal \(AC\) is twice the length of side \(AB\). Point \(K\) is chosen on side \(BC\) such that \(\angle KDB = \angle BDA\). Find the ratio \(BK : KC\).
|
2:1
|
olympiads
| 0.15625 |
A water tank has five pipes. To fill the tank, the first four pipes together take 6 hours; the last four pipes together take 8 hours. If only the first and fifth pipes are opened, it takes 12 hours to fill the tank. How many hours are needed to fill the tank if only the fifth pipe is used?
|
48
|
olympiads
| 0.328125 |
Given two linear functions $f(x)$ and $g(x)$ such that the graphs of $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. It is known that the graph of the function $y=(f(x))^{2}$ touches the graph of the function $y=-50g(x)$. Find all values of $A$ such that the graph of the function $y=(g(x))^{2}$ touches the graph of the function $y=\frac{f(x)}{A}$.
|
0.02
|
olympiads
| 0.109375 |
Find the integers \( n \) such that 5 divides \( 3n - 2 \) and 7 divides \( 2n + 1 \).
|
35k + 24
|
olympiads
| 0.140625 |
Let \([x]\) be the largest integer not greater than \(x\), for example, \([2.5] = 2\). If \(a = 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \cdots + \frac{1}{2004^{2}}\) and \(S = [a]\), find the value of \(a\).
|
1
|
olympiads
| 0.296875 |
Find all strictly positive integers \( x, y, p \) such that \( p^{x} - y^{p} = 1 \) where \( p \) is a prime number.
|
(1, 1, 2) \text{ and } (2, 2, 3)
|
olympiads
| 0.21875 |
A student made a calculation error in the final step by adding 20 instead of dividing by 20, resulting in an incorrect answer of 180. What should the correct answer be?
|
8
|
olympiads
| 0.09375 |
Around a round table there are 11 chairs placed regularly, and in front of each chair, there is a card with the name of one of the members of the JBMO committee. Each of the 11 cards has a different name. At the beginning of their meeting, the committee members realize that they all sat in front of a card that was not theirs. Is it possible to rotate the table so that at least 2 people end up in front of their own card?
|
\text{Yes, it is possible to turn the table such that at least 2 people find themselves in front of their own name cards.}
|
olympiads
| 0.234375 |
Find all values of \( n \), \( n \in \mathbb{N} \), for which the sum of the first \( n \) terms of the sequence \( a_k = 3k^2 - 3k + 1 \), \( k \in \mathbb{N} \), is equal to the sum of the first \( n \) terms of the sequence \( b_k = 2k + 89 \), \( k \in \mathbb{N} \). (12 points)
|
10
|
olympiads
| 0.359375 |
Let \( p \) be a given positive even number. The sum of all elements in the set \( A_{p}=\left\{x \mid 2^{p}<x<2^{p+1}, x=3m, m \in \mathbf{N}\} \) is \(\quad\).
|
2^{2p-1} - 2^{p-1}
|
olympiads
| 0.15625 |
If \(3 x^{3m+5n+9} + 4 y^{4m-2n-1} = 2\) is a linear equation in terms of \(x\) and \(y\), find the value of \(\frac{m}{n}\).
|
\frac{3}{19}
|
olympiads
| 0.203125 |
There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty boxes. How many boxes were there in total? Answer: 28182.
|
28182
|
olympiads
| 0.328125 |
Calculate the amount of personal income tax (НДФЛ) paid by Sergey over the past year if he is a resident of the Russian Federation and had a stable income of 30,000 rubles per month and a one-time vacation bonus of 20,000 rubles during this period. Last year, Sergey sold his car, which he inherited two years ago, for 250,000 rubles and bought a plot of land for building a house for 300,000 rubles. Sergey applied all applicable tax deductions. (Provide the answer without spaces and units of measurement.)
|
10400
|
olympiads
| 0.125 |
Find the locus of the midpoints of all chords of a given circle, which are equal to a given segment.
|
A concentric circle to the given circle
|
olympiads
| 0.0625 |
Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$ .
- $1\le x,y\le 29$
- $29\mid y^2-x^p-26$
|
2 \text{ and } 7
|
aops_forum
| 0.203125 |
Given that the side length of square $ABCD$ is 1, point $M$ is the midpoint of side $AD$, and with $M$ as the center and $AD$ as the diameter, a circle $\Gamma$ is drawn. Point $E$ is on segment $AB$, and line $CE$ is tangent to circle $\Gamma$. Find the area of $\triangle CBE$.
|
\frac{1}{4}
|
olympiads
| 0.09375 |
End of preview. Expand
in Data Studio
Default Math dataset used for https://github.com/RD211/actors.
The dataset is a subset of https://huggingface.co/datasets/SynthLabsAI/Big-Math-RL-Verified.
Only includes problems with solve rate between 0.05 and 0.6 and from aops_forum and olympiads.
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