problem stringlengths 3.65k 6.05k | answer stringlengths 1 5 | original_problem stringlengths 58 739 |
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# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 11 | Calculate the largest prime factor of $18^4 + 12^5 - 6^6$. |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 128 | Given that the focus of the parabola $y^{2}=8x$ is $F$, two perpendicular lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Let these lines intersect the parabola at points $A$, $B$ and $D$, $E$, respectively. The minimum area of quadrilateral $ADBE$ is ____. |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 1002 | How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$? |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 88 | An $8\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal? |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 32 | If 500 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 10 | A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two equal angles as indicated in the adjoining... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 40328 | Let $p(x)$ be a monic polynomial of degree 7 such that $p(0) = 0,$ $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ $p(6) = 6,$ and $p(7) = 7.$ Find $p(8).$ |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 2016 | Consider all 1000-element subsets of the set $\{1, 2, 3, \dots , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 13 | A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?
$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 8\sqrt{3} \qquad \textbf{(... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 85 | Given the equations $3x + 2y = 6$ and $2x + 3y = 7$, find $14x^2 + 25xy + 14y^2$. |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 4 | Call a triangle nice if the plane can be tiled using congruent copies of this triangle so that any two triangles that share an edge (or part of an edge) are reflections of each other via the shared edge. How many dissimilar nice triangles are there? |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 15 | Jack Sparrow needed to distribute 150 piastres into 10 purses. After putting some amount of piastres in the first purse, he placed more in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many pias... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 26 | Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.
[asy]/* AMC8 2002 #22 Problem */
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
draw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));
draw((1,0)--(1.5,0.5)--(1.5,1.5));
draw((0.5,1.5)--... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 1409 | Suppose \( a \) is an integer. A sequence \( x_1, x_2, x_3, x_4, \ldots \) is constructed with:
- \( x_1 = a \),
- \( x_{2k} = 2x_{2k-1} \) for every integer \( k \geq 1 \),
- \( x_{2k+1} = x_{2k} - 1 \) for every integer \( k \geq 1 \).
For example, if \( a = 2 \), then:
\[ x_1 = 2, \quad x_2 = 2x_1 = 4, \quad x_3 =... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 36 | Convex quadrilateral $A B C D$ has right angles $\angle A$ and $\angle C$ and is such that $A B=B C$ and $A D=C D$. The diagonals $A C$ and $B D$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $A M B$ and segment $C D$, respectively, such that points $P, M$, and $Q$ are collinear. Suppos... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 3000 | Five brothers divided their father's inheritance equally. The inheritance included three houses. Since it was not possible to split the houses, the three older brothers took the houses, and the younger brothers were given money: each of the three older brothers paid $2,000. How much did one house cost in dollars? |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 36 | Given that there are 5 advertisements (3 commercial advertisements and 2 Olympic promotional advertisements), the last advertisement is an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively, determine the number of different broadcasting methods. |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 30 | A point $Q$ is chosen inside $\triangle DEF$ such that lines drawn through $Q$ parallel to $\triangle DEF$'s sides decompose it into three smaller triangles $u_1$, $u_2$, and $u_3$, which have areas $3$, $12$, and $15$ respectively. Determine the area of $\triangle DEF$. |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 990 | Given that $x$ is a multiple of $46200$, determine the greatest common divisor of $f(x) = (3x + 5)(5x + 3)(11x + 6)(x + 11)$ and $x$. |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 677 | Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2 = \sqrt{11}$. Find $\sum_{k=1}^4(CE_k)^2$. |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 43 | Grace has $\$4.80$ in U.S. coins. She has the same number of dimes and pennies. What is the greatest number of dimes she could have? |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 82310 | A mathematician $M^{\prime}$ is called a descendent of mathematician $M$ if there is a sequence of mathematicians $M=M_{1}, M_{2}, \ldots, M_{k}=M^{\prime}$ such that $M_{i}$ was $M_{i+1}$ 's doctoral advisor for all $i$. Estimate the number of descendents that the mathematician who has had the largest number of descen... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 4002 | Determine the value of $$2002+\frac{1}{2}\left(2001+\frac{1}{2}\left(2000+\cdots+\frac{1}{2}\left(3+\frac{1}{2} \cdot 2\right)\right) \cdots\right)$$ |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 6 | How many different routes are there from point $A$ to point $B$ if you can only move to the right or down along the drawn segments?
[asy]
unitsize(0.09inch);
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((5,0)--(5,10));
draw((0,5)--(10,5));
dot((0,10));
dot((10,0));
label("$A$",(0,10),NW);
label("$B$",(10,0),SE);
... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 2 | For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 5049 | A convex 2019-gon \(A_{1}A_{2}\ldots A_{2019}\) is cut into smaller pieces along its 2019 diagonals of the form \(A_{i}A_{i+3}\) for \(1 \leq i \leq 2019\), where \(A_{2020}=A_{1}, A_{2021}=A_{2}\), and \(A_{2022}=A_{3}\). What is the least possible number of resulting pieces? |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 999 | Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
Find the value of \( f(90) \). |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 17 | Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{4}$ and $\frac{BY}{BD} = \frac{11}{36}$.
Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Le... |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 492 | Given a 50-term sequence $(b_1, b_2, \dots, b_{50})$, the Cesaro sum is 500. What is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \dots, b_{50})$? |
# GENERATOR (PROBLEM SOLVER)
Instruction: You are an expert problem-solving assistant tasked with analyzing and solving various questions using a combination of your expertise and provided reference materials. Each task will include:
1. A specific question or problem to solve
2. A cheatsheet containing relevant strate... | 216 | Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exact... |
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