problem stringlengths 100 3.44k | answer stringlengths 0 109 | task stringclasses 6
values | problem_number int64 8 1.31k | set_number int64 1 5 |
|---|---|---|---|---|
There exist real numbers $x$ and $y$, both greater than 1, such that $\log_x\left(y^x\right)=\log_y\left(x^{4y}\right)=10$. Find $xy$. | 025 | AIME24 | 8 | 2 |
Let \(b\ge 2\) be an integer. Call a positive integer \(n\) \(b\text-\textit{eautiful}\) if it has exactly two digits when expressed in base \(b\) and these two digits sum to \(\sqrt n\). For example, \(81\) is \(13\text-\textit{eautiful}\) because \(81 = \underline{6} \ \underline{3}_{13} \) and \(6 + 3 = \sqrt{81}... | 211 | AIME24 | 21 | 1 |
The $9$ members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was gr... | 16 | AIME25 | 33 | 5 |
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$ | 468 | AIME25 | 46 | 4 |
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such\[AK = AL = BK = CL = KL = 14.\]The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$ | 104 | AIME25 | 59 | 3 |
If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\ov... | 29 | AMC23 | 72 | 2 |
Calculate the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m^2+n^2$? | 13 | AMC23 | 85 | 1 |
When the roots of the polynomial
\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\]
are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive? | 6 | AMC23 | 97 | 5 |
Sigmatropic rearrangement is a class of pericyclic reactions. In these reactions, the terminal pi bond migrates into the sigma bond. Cope and Claisen rearrangements are also sigma tropic rearrangements. These reactions are usually thermodynamically favored. Complete the following reactions.
1,1-dimethoxyethan-1-amine +... | A = (Z)-1-(but-2-en-2-yloxy)ethen-1-amine, B = (3Z,4E)-3,4-diethylidenecyclobut-1-ene, C = 4-methylenehexanal | GPQADiamond | 149 | 1 |
You have prepared an unknown compound. To identify the product, you have used the following characterisation techniques: FTIR, 1H NMR, and Mass Spectrometry. The FTIR spectrum shows a very broad absorption peak at 3000 wavenumbers. Two other strong absorption peaks are observed at 1700 and 1650 wavenumbers. Several pea... | C6H10O2 | GPQADiamond | 213 | 1 |
Astronomers are interested in the luminosities of two stars. Star_1 has a radius 1.5 times larger than that of Star_2. The mass of Star_1 is also larger by the same factor (1.5 times). Through spectroscopy, the astronomers managed to determine the wavelengths at which the stars appeared brightest, and found that they a... | ~2.23
| GPQADiamond | 277 | 1 |
The solution to $-4 < 2(x - 1) < 8$ is expressed in the form $a < x < b$. Find the value of $a + b$. | 4 | MATH500 | 341 | 1 |
At what value of $y$ is there a horizontal asymptote for the graph of the equation $y=\frac{4x^3+2x-4}{3x^3-2x^2+5x-1}$? | \frac43 | MATH500 | 405 | 1 |
Find the product of the $y$-coordinates of all the distinct solutions $(x,y)$ for the two equations $y=x^2-8$ and $y^2=-5x+44$. | 1736 | MATH500 | 469 | 1 |
Let $f$ be the function defined by $f(x) = x^3 - 49x^2 + 623x - 2015,$ and let $g(x) = f(x + 5).$ Compute the sum of the roots of $g.$ | 34 | MATH500 | 533 | 1 |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be three vectors such that
\[\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix}, \quad \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix}, \quad \mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix}.\]Compute ... | \begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix} | MATH500 | 597 | 1 |
Two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president. Each candidate has a 50$\%$ chance of winning. What is the probability that both Dan and Freddie win? (Express your answer as a fraction.) | \frac14 | MATH500 | 661 | 1 |
A square and an equilateral triangle have equal perimeters. The area of the triangle is $16\sqrt{3}$ square centimeters. How long, in centimeters, is a diagonal of the square? Express your answer in simplest radical form.
[asy]
defaultpen(linewidth(1));
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
pair a = (1.25,0)+1.25*d... | 6\sqrt{2} | MATH500 | 725 | 1 |
Suppose the polynomial
$$f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$$has integer coefficients, and its roots are distinct integers.
Given that $a_n=2$ and $a_0=66$, what is the least possible value of $|a_{n-1}|$? | 14 | MATH500 | 789 | 1 |
You are given a 0-indexed integer array nums representing the strength of some heroes. The power of a group of heroes is defined as follows:
Let i_0, i_1, ... ,i_k be the indices of the heroes in a group. Then, the power of this group is max(nums[i_0], nums[i_1], ... ,nums[i_k])^2 * min(nums[i_0], nums[i_1], ... ,nums... | LiveCodeBench | 817 | 1 | |
You are given a 0-indexed integer array nums representing the initial positions of some marbles. You are also given two 0-indexed integer arrays moveFrom and moveTo of equal length.
Throughout moveFrom.length steps, you will change the positions of the marbles. On the i^th step, you will move all marbles at position mo... | LiveCodeBench | 838 | 2 | |
Given three strings a, b, and c, your task is to find a string that has the minimum length and contains all three strings as substrings.
If there are multiple such strings, return the lexicographically smallest one.
Return a string denoting the answer to the problem.
Notes
A string a is lexicographically smaller than ... | LiveCodeBench | 859 | 3 | |
You are given an integer array nums and two positive integers m and k.
Return the maximum sum out of all almost unique subarrays of length k of nums. If no such subarray exists, return 0.
A subarray of nums is almost unique if it contains at least m distinct elements.
A subarray is a contiguous non-empty sequence of el... | LiveCodeBench | 881 | 1 | |
You are given an array nums consisting of non-negative integers.
We define the score of subarray nums[l..r] such that l <= r as nums[l] AND nums[l + 1] AND ... AND nums[r] where AND is the bitwise AND operation.
Consider splitting the array into one or more subarrays such that the following conditions are satisfied:
E... | LiveCodeBench | 902 | 2 | |
You are given a 0-indexed integer array nums.
A subsequence of nums having length k and consisting of indices i_0 < i_1 < ... < i_k-1 is balanced if the following holds:
nums[i_j] - nums[i_j-1] >= i_j - i_j-1, for every j in the range [1, k - 1].
A subsequence of nums having length 1 is considered balanced.
Return an... | LiveCodeBench | 923 | 3 | |
You are given an integer n and a 0-indexed integer array sick which is sorted in increasing order.
There are n children standing in a queue with positions 0 to n - 1 assigned to them. The array sick contains the positions of the children who are infected with an infectious disease. An infected child at position i can s... | LiveCodeBench | 945 | 1 | |
You are given a 0-indexed array of positive integers nums.
A subarray of nums is called incremovable if nums becomes strictly increasing on removing the subarray. For example, the subarray [3, 4] is an incremovable subarray of [5, 3, 4, 6, 7] because removing this subarray changes the array [5, 3, 4, 6, 7] to [5, 6, 7]... | LiveCodeBench | 966 | 2 | |
You are given a 0-indexed integer array nums, and an integer k.
In one operation, you can remove one occurrence of the smallest element of nums.
Return the minimum number of operations needed so that all elements of the array are greater than or equal to k.
Example 1:
Input: nums = [2,11,10,1,3], k = 10
Output: 3
Ex... | LiveCodeBench | 987 | 3 | |
There is a rectangular cake with some strawberries on the xy-plane. The cake occupies the rectangular area \lbrace (x, y) : 0 \leq x \leq W, 0 \leq y \leq H \rbrace.
There are N strawberries on the cake, and the coordinates of the i-th strawberry are (p_i, q_i) for i = 1, 2, \ldots, N. No two strawberries have the same... | LiveCodeBench | 1,009 | 1 | |
You are given a length-N sequence A=(A_1,A_2,\dots,A_N) consisting of 0, 1, and 2,
and a length-N string S=S_1S_2\dots S_N consisting of M, E, and X.
Find the sum of
\text{mex}(A_i,A_j,A_k) over all tuples of integers (i,j,k) such that 1 \leq i < j < k \leq N and S_iS_jS_k= MEX.
Here, \text{mex}(A_i,A_j,A_k) denotes... | LiveCodeBench | 1,030 | 2 | |
You are given a non-empty string S consisting of (, ), and ?.
There are 2^x ways to obtain a new string by replacing each ? in S with ( and ), where x is the number of occurrences of ? in S. Among them, find the number, modulo 998244353, of ways that yield a parenthesis string.
A string is said to be a parenthesis st... | LiveCodeBench | 1,051 | 3 | |
You are given a sequence of positive integers of length N: A=(A_1,A_2,\ldots,A_N). Find the number of triples of positive integers (i,j,k) that satisfy all of the following conditions:
- 1\leq i < j < k\leq N,
- A_i = A_k,
- A_i \neq A_j.
Input
The input is given from Standard Input in the following format:
N
A_1... | LiveCodeBench | 1,073 | 1 | |
There are N players numbered 1 to N, who have played a round-robin tournament. For every match in this tournament, one player won and the other lost.
The results of the matches are given as N strings S_1,S_2,\ldots,S_N of length N each, in the following format:
-
If i\neq j, the j-th character of S_i is o or x. o mea... | LiveCodeBench | 1,094 | 2 | |
You are given a string S of length N consisting of lowercase English letters.
If there are any adjacent occurrences of a and b in S, print Yes; otherwise, print No. (The order of a and b does not matter.)
Input
The input is given from Standard Input in the following format:
N
S
Output
If there are any adjacent oc... | LiveCodeBench | 1,115 | 3 | |
A supermarket sells egg packs.
A pack of 6 eggs costs S yen, a pack of 8 eggs costs M yen, and a pack of 12 eggs costs L yen.
When you can buy any number of each pack, find the minimum amount of money required to purchase at least N eggs.
Input
The input is given from Standard Input in the following format:
N S M L
... | LiveCodeBench | 1,137 | 1 | |
There is a grid with N rows and N columns, where N is an odd number at most 45.
Let (i,j) denote the cell at the i-th row from the top and j-th column from the left.
In this grid, you will place Takahashi and a dragon consisting of N^2-1 parts numbered 1 to N^2-1 in such a way that satisfies the following conditions:
... | LiveCodeBench | 1,158 | 2 | |
You have an empty sequence A. There are Q queries given, and you need to process them in the order they are given.
The queries are of the following two types:
- 1 x: Append x to the end of A.
- 2 k: Find the k-th value from the end of A. It is guaranteed that the length of A is at least k when this query is given.
In... | LiveCodeBench | 1,179 | 3 | |
You are given three sequences A=(A_1,\ldots,A_N), B=(B_1,\ldots,B_M), and C=(C_1,\ldots,C_L).
Additionally, a sequence X=(X_1,\ldots,X_Q) is given. For each i=1,\ldots,Q, solve the following problem:
Problem: Is it possible to select one element from each of A, B, and C so that their sum is X_i?
Input
The input is gi... | LiveCodeBench | 1,201 | 1 | |
A string S consisting of lowercase English letters is a good string if and only if it satisfies the following property for all integers i not less than 1:
- There are exactly zero or exactly two different letters that appear exactly i times in S.
Given a string S, determine if it is a good string.
Input
The input i... | LiveCodeBench | 1,222 | 2 | |
There are N buildings aligned in a row. The i-th building from the left has a height of H_i.
Determine if there is a building taller than the first one from the left. If such a building exists, find the position of the leftmost such building from the left.
Input
The input is given from Standard Input in the following... | LiveCodeBench | 1,243 | 3 | |
You are given a positive integer k. Initially, you have an array nums = [1].
You can perform any of the following operations on the array any number of times (possibly zero):
Choose any element in the array and increase its value by 1.
Duplicate any element in the array and add it to the end of the array.
Return the ... | LiveCodeBench | 1,265 | 1 | |
You are given an array of positive integers nums.
Return the number of subarrays of nums, where the first and the last elements of the subarray are equal to the largest element in the subarray.
Example 1:
Input: nums = [1,4,3,3,2]
Output: 6
Explanation:
There are 6 subarrays which have the first and the last element... | LiveCodeBench | 1,286 | 2 | |
A powerful array for an integer x is the shortest sorted array of powers of two that sum up to x. For example, the powerful array for 11 is [1, 2, 8].
The array big_nums is created by concatenating the powerful arrays for every positive integer i in ascending order: 1, 2, 3, and so forth. Thus, big_nums starts as [1, 2... | LiveCodeBench | 1,307 | 3 |
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