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problem
stringlengths 25
2.73k
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stringlengths 0
1.22k
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stringclasses 5
values | difficulty
float64 0
0
| gemini_solution_reward
int64 1
1
| gemini_solution
listlengths 4
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| student_solutions
listlengths 32
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"In triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point(...TRUNCATED) |
34
|
dapo
| 0 | 1 | [{"content":"In triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D(...TRUNCATED) | ["We are given a triangle $ ABC $ with:\n\n- $ \\sin \\angle A = \\frac{4}{5} $, and $ \\angle A < 9(...TRUNCATED) |
"Determine the smallest positive integer $A$ with an odd number of digits such that both $A$ and the(...TRUNCATED) |
100902018
|
dapo
| 0 | 1 | [{"content":"Determine the smallest positive integer $A$ with an odd number of digits such that both(...TRUNCATED) | ["We are to find the **smallest positive integer $ A $** with an **odd number of digits** such that:(...TRUNCATED) |
"Find the smallest integer $n > 3$ such that, for each partition of $\\{3, 4, \\ldots, n\\}$ into tw(...TRUNCATED) |
243
|
dapo
| 0 | 1 | [{"content":"Find the smallest integer $n > 3$ such that, for each partition of $\\{3, 4, \\ldots, n(...TRUNCATED) | ["We are to find the **smallest integer $ n > 3 $** such that **every partition** of the set $ \\{3,(...TRUNCATED) |
"Find all the ways of placing the integers $1, 2, 3, \\ldots, 16$ in the boxes below, such that each(...TRUNCATED) |
1
|
dapo
| 0 | 1 | [{"content":"Find all the ways of placing the integers $1, 2, 3, \\ldots, 16$ in the boxes below, su(...TRUNCATED) | ["We are given a sequence of 16 boxes (in a line), and we are to place the integers $1, 2, 3, \\ldot(...TRUNCATED) |
"Compute the smallest integer $n \\geq 4$ such that $\\binom{n}{4}$ ends in 4 or more zeroes (i.e., (...TRUNCATED) |
8128
|
dapo
| 0 | 1 | [{"content":"Compute the smallest integer $n \\geq 4$ such that $\\binom{n}{4}$ ends in 4 or more ze(...TRUNCATED) | ["We are asked to find the **smallest integer $ n \\geq 4 $** such that \n$$\n\\binom{n}{4}\n$$\nen(...TRUNCATED) |
"Define a *growing spiral* in the plane as a sequence of points with integer coordinates $P_0=(0,0),(...TRUNCATED) |
10053
|
dapo
| 0 | 1 | [{"content":"Define a *growing spiral* in the plane as a sequence of points with integer coordinates(...TRUNCATED) | ["We are given a definition of a **growing spiral** and asked to determine how many lattice points $(...TRUNCATED) |
What is the maximal number of crosses that can fit in a $10 \times 11$ board without overlapping?
|
14
|
dapo
| 0 | 1 | [{"content":"What is the maximal number of crosses that can fit in a $10 \\times 11$ board without o(...TRUNCATED) | ["We are asked to find the **maximal number of crosses** that can fit on a $10 \\times 11$ board **w(...TRUNCATED) |
"A large number of rocks are placed on a table. On each turn, one may remove some rocks from the tab(...TRUNCATED) |
18
|
dapo
| 0 | 1 | [{"content":"A large number of rocks are placed on a table. On each turn, one may remove some rocks (...TRUNCATED) | ["We are given a game where rocks are removed from a table following specific rules:\n\n- On the **f(...TRUNCATED) |
"Does there exist a four-digit positive integer with different non-zero digits, which has the follow(...TRUNCATED) |
1234
|
dapo
| 0 | 1 | [{"content":"Does there exist a four-digit positive integer with different non-zero digits, which ha(...TRUNCATED) | ["We are asked whether there exists a **four-digit positive integer** with **different non-zero digi(...TRUNCATED) |
"The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. Th(...TRUNCATED) |
60
|
dapo
| 0 | 1 | [{"content":"The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, (...TRUNCATED) | ["We are given a target made up of **concentric circles** with diameters:\n\n- 4, 8, 12, 16, and 20.(...TRUNCATED) |
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