problem stringlengths 95 3.07k | answer stringlengths 0 48 | task stringclasses 6
values | problem_number int64 6 1.31k | set_number int64 1 5 |
|---|---|---|---|---|
Let $ABCD$ be a tetrahedron such that $AB=CD= \sqrt{41}$, $AC=BD= \sqrt{80}$, and $BC=AD= \sqrt{89}$. There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m \sqrt n}{p}$, where $m$, $n$, an... | 104 | AIME24 | 6 | 5 |
Let \(O=(0,0)\), \(A=\left(\tfrac{1}{2},0\right)\), and \(B=\left(0,\tfrac{\sqrt{3}}{2}\right)\) be points in the coordinate plane. Let \(\mathcal{F}\) be the family of segments \(\overline{PQ}\) of unit length lying in the first quadrant with \(P\) on the \(x\)-axis and \(Q\) on the \(y\)-axis. There is a unique point... | 023 | AIME24 | 19 | 4 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ thro... | 588 | AIME25 | 32 | 3 |
Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$. | 735 | AIME25 | 45 | 2 |
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]$x_{2025}$ can be expressed as $ rac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$. | 248 | AIME25 | 58 | 1 |
The weight of $\frac{1}{3}$ of a large pizza together with $3 \frac{1}{2}$ cups of orange slices is the same as the weight of $\frac{3}{4}$ of a large pizza together with $\frac{1}{2}$ cup of orange slices. A cup of orange slices weighs $\frac{1}{4}$ of a pound. What is the weight, in pounds, of a large pizza? The answ... | 4 | AMC23 | 70 | 5 |
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four... | 7 | AMC23 | 83 | 4 |
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a ... | 4 | AMC23 | 96 | 3 |
Astronomers are studying a system of three exoplanets (Planet1, Planet2, and Planet3) with circular orbits discovered through the TTV method. They have found that the ratio of the equilibrium temperatures between Planet1 and Planet2 is approximately 1.4, and between Planet2 and Planet3, it is about 2.3. They have also ... | ~ 33.4 | GPQADiamond | 142 | 1 |
The reaction between cyclohexanone and piperidine can result in the formation of a cyclic imine, also known as a Schiff base. This type of reaction is a classic example of imine formation, which involves the condensation of a ketone (cyclohexanone in this case) with a primary amine (piperidine) in the presence of an ac... | A = TsOH, B = 3-(2-oxocyclohexyl)propanal | GPQADiamond | 206 | 1 |
An electron is in the spin state (3i, 4). Find the expectation value of its spin along y-direction, S_y.
Note: \sigma_y (in latex format) is:
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix} | -12*hbar/25
| GPQADiamond | 270 | 1 |
If $n \equiv 2 \pmod{7}$, then find the remainder when $(n + 2)(n + 4)(n + 6)$ is divided by 7. | 3 | MATH500 | 334 | 1 |
If $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},$ then find the vector $\mathbf{v}$ such that $\mathbf{a} \cdot \mathbf{v} = 2$ and $\mathbf{a} \times \mathbf{v} = \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}.$ | \begin{pmatrix} -1/3 \\ 2/3 \\ 5/3 \end{pmatrix} | MATH500 | 398 | 1 |
If $-6\leq a \leq -2$ and $3 \leq b \leq 5$, what is the greatest possible value of $\displaystyle\left(a+\frac{1}{b}\right)\left(\frac{1}{b}-a\right) $? Express your answer as a common fraction. | -\frac{35}{9} | MATH500 | 462 | 1 |
For a constant $c,$ in cylindrical coordinates $(r,\theta,z),$ find the shape described by the equation
\[z = c.\](A) Line
(B) Circle
(C) Plane
(D) Sphere
(E) Cylinder
(F) Cone
Enter the letter of the correct option. | \text{(C)} | MATH500 | 526 | 1 |
Consider the rectangular region with the following points as vertices: $$(5,4), (-5,4), (-5,-4), (5,-4).$$ How many points with integer coordinates will be strictly in the interior of this rectangular region? | 63 | MATH500 | 590 | 1 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[\frac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc}.\] | 100 | MATH500 | 654 | 1 |
There are an infinite number of vectors $\mathbf{v}$ such that
\[\begin{pmatrix} 1 \\ 2 \\ -5 \end{pmatrix} \times \mathbf{v} = \begin{pmatrix} 90 \\ 30 \\ 30 \end{pmatrix}.\]Find the vector $\mathbf{v}$ that has the smallest magnitude. | \begin{pmatrix} -7 \\ 16 \\ 5 \end{pmatrix} | MATH500 | 718 | 1 |
The area of triangle $ABC$ is equal to $a^2 - (b - c)^2,$ where $a,$ $b,$ and $c$ are the sides of triangle $ABC,$ as usual. Compute $\tan A.$ | \frac{8}{15} | MATH500 | 782 | 1 |
You are given two numeric strings num1 and num2 and two integers max_sum and min_sum. We denote an integer x to be good if:
num1 <= x <= num2
min_sum <= digit_sum(x) <= max_sum.
Return the number of good integers. Since the answer may be large, return it modulo 10^9 + 7.
Note that digit_sum(x) denotes the sum of the ... | LiveCodeBench | 814 | 3 | |
You are given a 0-indexed integer array nums and an integer k.
A subarray is called equal if all of its elements are equal. Note that the empty subarray is an equal subarray.
Return the length of the longest possible equal subarray after deleting at most k elements from nums.
A subarray is a contiguous, possibly empty ... | LiveCodeBench | 836 | 1 | |
You are given an integer n. We say that two integers x and y form a prime number pair if:
1 <= x <= y <= n
x + y == n
x and y are prime numbers
Return the 2D sorted list of prime number pairs [x_i, y_i]. The list should be sorted in increasing order of x_i. If there are no prime number pairs at all, return an empty a... | LiveCodeBench | 857 | 2 | |
Given two positive integers low and high represented as strings, find the count of stepping numbers in the inclusive range [low, high].
A stepping number is an integer such that all of its adjacent digits have an absolute difference of exactly 1.
Return an integer denoting the count of stepping numbers in the inclusive... | LiveCodeBench | 878 | 3 | |
You are given a 1-indexed array nums of n integers.
A set of numbers is complete if the product of every pair of its elements is a perfect square.
For a subset of the indices set {1, 2, ..., n} represented as {i_1, i_2, ..., i_k}, we define its element-sum as: nums[i_1] + nums[i_2] + ... + nums[i_k].
Return the maximum... | LiveCodeBench | 900 | 1 | |
You are given a 0-indexed array nums of integers.
A triplet of indices (i, j, k) is a mountain if:
i < j < k
nums[i] < nums[j] and nums[k] < nums[j]
Return the minimum possible sum of a mountain triplet of nums. If no such triplet exists, return -1.
Example 1:
Input: nums = [8,6,1,5,3]
Output: 9
Explanation: Tripl... | LiveCodeBench | 921 | 2 | |
You are given a 0-indexed integer array batteryPercentages having length n, denoting the battery percentages of n 0-indexed devices.
Your task is to test each device i in order from 0 to n - 1, by performing the following test operations:
If batteryPercentages[i] is greater than 0:
Increment the count of tested dev... | LiveCodeBench | 942 | 3 | |
You are given a 0-indexed integer array nums and a positive integer k.
You can apply the following operation on the array any number of times:
Choose any element of the array and flip a bit in its binary representation. Flipping a bit means changing a 0 to 1 or vice versa.
Return the minimum number of operations requ... | LiveCodeBench | 964 | 1 | |
You are given an integer array nums of even length. You have to split the array into two parts nums1 and nums2 such that:
nums1.length == nums2.length == nums.length / 2.
nums1 should contain distinct elements.
nums2 should also contain distinct elements.
Return true if it is possible to split the array, and false ot... | LiveCodeBench | 985 | 2 | |
There are N people numbered 1, 2, \ldots, N, sitting in this clockwise order around a round table.
In particular, person 1 is sitting next to person N in the clockwise direction.
For each i = 1, 2, \ldots, N, person i has a name S_i and an age A_i.
Here, no two people have the same name or the same age.
Starting from... | LiveCodeBench | 1,006 | 3 | |
N people numbered 1 through N tossed a coin several times. We know that person i's tosses resulted in A_i heads and B_i tails.
Person i's success rate of the tosses is defined by \displaystyle\frac{A_i}{A_i+B_i}. Sort people 1,\ldots,N in descending order of their success rates, with ties broken in ascending order of... | LiveCodeBench | 1,028 | 1 | |
Takahashi invented Tak Code, a two-dimensional code. A TaK Code satisfies all of the following conditions:
- It is a region consisting of nine horizontal rows and nine vertical columns.
- All the 18 cells in the top-left and bottom-right three-by-three regions are black.
- All the 14 cells that are adjacent (horizont... | LiveCodeBench | 1,049 | 2 | |
There are N rectangular sheets spread out on a coordinate plane.
Each side of the rectangular region covered by each sheet is parallel to the x- or y-axis.
Specifically, the i-th sheet covers exactly the region satisfying A_i \leq x\leq B_i and C_i \leq y\leq D_i.
Let S be the area of the region covered by one or more... | LiveCodeBench | 1,070 | 3 | |
AtCoder Inc. is planning to develop a product. The product has K parameters, whose values are currently all zero. The company aims to raise all parameter values to at least P.
There are N development plans. Executing the i-th development plan (1 \le i \le N) increases the value of the j-th parameter by A_{i,j} for ever... | LiveCodeBench | 1,092 | 1 | |
You are given an integer N and strings R and C of length N consisting of A, B, and C. Solve the following problem.
There is a N \times N grid. All cells are initially empty.
You can write at most one character from A, B, and C in each cell. (You can also leave the cell empty.)
Determine if it is possible to satisfy al... | LiveCodeBench | 1,113 | 2 | |
You are given an N \times N grid. Let (i,j) denote the cell in the i-th row from the top and the j-th column from the left.
The states of the cells are given by N strings of length N, S_1, S_2, \dots, S_N, in the following format:
- If the j-th character of S_i is o, there is an o written in cell (i,j).
- If the j-th... | LiveCodeBench | 1,134 | 3 | |
You are given an integer N.
Print all triples of non-negative integers (x,y,z) such that x+y+z\leq N in ascending lexicographical order.
What is lexicographical order for non-negative integer triples?
A triple of non-negative integers (x,y,z) is said to be lexicographically smaller than (x',y',z') if and only if one... | LiveCodeBench | 1,156 | 1 | |
There is an N \times N grid, where each cell is either empty or contains an obstacle. Let (i, j) denote the cell at the i-th row from the top and the j-th column from the left.
There are also two players on distinct empty cells of the grid. The information about each cell is given as N strings S_1, S_2, \ldots, S_N of ... | LiveCodeBench | 1,177 | 2 | |
In a coordinate space, we want to place three cubes with a side length of 7 so that the volumes of the regions contained in exactly one, two, three cube(s) are V_1, V_2, V_3, respectively.
For three integers a, b, c, let C(a,b,c) denote the cubic region represented by (a\leq x\leq a+7) \land (b\leq y\leq b+7) \land (c... | LiveCodeBench | 1,198 | 3 | |
A string T of length 3 consisting of uppercase English letters is an airport code for a string S of lowercase English letters if and only if T can be derived from S by one of the following methods:
- Take a subsequence of length 3 from S (not necessarily contiguous) and convert it to uppercase letters to form T.
- Tak... | LiveCodeBench | 1,220 | 1 | |
For positive integers x and y, define f(x, y) as the remainder of (x + y) divided by 10^8.
You are given a sequence of positive integers A = (A_1, \ldots, A_N) of length N. Find the value of the following expression:
\displaystyle \sum_{i=1}^{N-1}\sum_{j=i+1}^N f(A_i,A_j).
Input
The input is given from Standard Input... | LiveCodeBench | 1,241 | 2 | |
You are given a string word and an integer k.
We consider word to be k-special if |freq(word[i]) - freq(word[j])| <= k for all indices i and j in the string.
Here, freq(x) denotes the frequency of the character x in word, and |y| denotes the absolute value of y.
Return the minimum number of characters you need to delet... | LiveCodeBench | 1,262 | 3 | |
You are given an integer array nums of length n, and a positive integer k.
The power of a subsequence is defined as the minimum absolute difference between any two elements in the subsequence.
Return the sum of powers of all subsequences of nums which have length equal to k.
Since the answer may be large, return it mod... | LiveCodeBench | 1,284 | 1 | |
You are given an array nums consisting of integers. You are also given a 2D array queries, where queries[i] = [pos_i, x_i].
For query i, we first set nums[pos_i] equal to x_i, then we calculate the answer to query i which is the maximum sum of a subsequence of nums where no two adjacent elements are selected.
Return th... | LiveCodeBench | 1,305 | 2 |
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