problem stringlengths 98 2.44k | answer stringlengths 0 44 | 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 | 5 |
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 | 4 |
Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$. | 117 | AIME25 | 34 | 3 |
Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$. | 49 | AIME25 | 47 | 2 |
Let\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$. | 240 | AIME25 | 60 | 1 |
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 | 5 |
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 | 4 |
For how many integers $n$ does the expression\[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}\]represent a real number, where log denotes the base $10$ logarithm? | 901 | AMC23 | 98 | 3 |
Astronomers are studying a star with a 1.5 solar radius and 1.1 solar masses. When the star's surface is not covered by dark spots, its Teff is 6000K. However, when 40% of its surface is covered by spots, the overall photospheric effective temperature decreases to 5500 K. In the stellar photosphere, when examining the ... | ~4.5 | GPQADiamond | 152 | 1 |
Compound A shows two signals in the proton NMR, a triplet at 0.9 ppm for 6H and a quartet at 1.3 ppm for 4H. It undergoes monobromination to form compound B. B reacts with alcoholic KOH to form C, which has two geometrical isomers. The cis-isomer of C reacts with (1E,3E)-penta-1,3-dien-1-ol to produce compound D. Ident... | (1S,4R,5S,6S)-4,5,6-trimethylcyclohex-2-enol | GPQADiamond | 216 | 1 |
Consider an isolated system of 13 identical particles each with charge 2e and negligible mass. 12 of these charges are
constrained to stay at 2 m from a point P. The 13th charge is fixed at P. What is the minimum energy (in Joules) of this system
correct to three decimals? | 2.822 x 10^-26 | GPQADiamond | 280 | 1 |
In right triangle $ABC$ with $\angle B = 90^\circ$, we have $\sin A = 2\cos A$. What is $\tan A$? | 2 | MATH500 | 344 | 1 |
Find the degree measure of the least positive angle $\theta$ for which
\[\tan \theta = \frac{\cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ - \sin 5^\circ \sin 20^\circ - \sin 35^\circ \sin 50^\circ}{\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^... | 120^\circ | MATH500 | 408 | 1 |
Consider the geometric series $4+\frac{12}{a}+\frac{36}{a^2}+\cdots$. If the sum is a perfect square, what is the smallest possible value of $a$ where $a$ is a positive integer? | 4 | MATH500 | 472 | 1 |
If $x$, $y$, and $z$ are positive with $xy = 24$, $xz = 48$, and $yz = 72$, then find $x + y + z.$ | 22 | MATH500 | 536 | 1 |
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American countries an... | 5.4 \text{ cents} | MATH500 | 600 | 1 |
If $a$ and $b$ are positive integers such that $\gcd(a,b)=210$, $\mathop{\text{lcm}}[a,b]=210^3$, and $a<b$, how many possible values are there for $a$? | 8 | MATH500 | 664 | 1 |
Let $a$ and $b$ be positive real numbers such that $ab^2 = 5.$ Find the minimum value of $a^4 + b^8.$ | 50 | MATH500 | 728 | 1 |
The number $(\sqrt{2}+\sqrt{3})^3$ can be written in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$, where $a$, $b$, and $c$ are integers. What is $a+b+c$? | 20 | MATH500 | 792 | 1 |
You are given a 0-indexed permutation of n integers nums.
A permutation is called semi-ordered if the first number equals 1 and the last number equals n. You can perform the below operation as many times as you want until you make nums a semi-ordered permutation:
Pick two adjacent elements in nums, then swap them.
Re... | LiveCodeBench | 818 | 1 | |
You are given two integers num1 and num2.
In one operation, you can choose integer i in the range [0, 60] and subtract 2^i + num2 from num1.
Return the integer denoting the minimum number of operations needed to make num1 equal to 0.
If it is impossible to make num1 equal to 0, return -1.
Example 1:
Input: num1 = 3,... | LiveCodeBench | 839 | 2 | |
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 subarray of size k from the array and decrease all its elements by 1.
Return true if you can make all the array elements equal to 0, or false otherwise.
A subarray ... | LiveCodeBench | 860 | 3 | |
Initially, you have a bank account balance of 100 dollars.
You are given an integer purchaseAmount representing the amount you will spend on a purchase in dollars.
At the store where you will make the purchase, the purchase amount is rounded to the nearest multiple of 10. In other words, you pay a non-negative amount, ... | LiveCodeBench | 882 | 1 | |
You are given a 0-indexed sorted array of integers nums.
You can perform the following operation any number of times:
Choose two indices, i and j, where i < j, such that nums[i] < nums[j].
Then, remove the elements at indices i and j from nums. The remaining elements retain their original order, and the array is re-in... | LiveCodeBench | 903 | 2 | |
There are n teams numbered from 0 to n - 1 in a tournament.
Given a 0-indexed 2D boolean matrix grid of size n * n. For all i, j that 0 <= i, j <= n - 1 and i != j team i is stronger than team j if grid[i][j] == 1, otherwise, team j is stronger than team i.
Team a will be the champion of the tournament if there is no t... | LiveCodeBench | 924 | 3 | |
You are given an integer array nums and an integer k.
The frequency of an element x is the number of times it occurs in an array.
An array is called good if the frequency of each element in this array is less than or equal to k.
Return the length of the longest good subarray of nums.
A subarray is a contiguous non-empt... | LiveCodeBench | 946 | 1 | |
You are given a 0-indexed integer array nums and an integer k.
In one operation, you can pick any index i of nums such that 0 <= i < nums.length - 1 and replace nums[i] and nums[i + 1] with a single occurrence of nums[i] & nums[i + 1], where & represents the bitwise AND operator.
Return the minimum possible value of th... | LiveCodeBench | 967 | 2 | |
You are given a 1-indexed array of distinct integers nums of length n.
You need to distribute all the elements of nums between two arrays arr1 and arr2 using n operations. In the first operation, append nums[1] to arr1. In the second operation, append nums[2] to arr2. Afterwards, in the i^th operation:
If the last ele... | LiveCodeBench | 988 | 3 | |
You are given an undirected graph G with N vertices and M edges.
For i = 1, 2, \ldots, M, the i-th edge is an undirected edge connecting vertices u_i and v_i.
A graph with N vertices is called good if the following condition holds for all i = 1, 2, \ldots, K:
- there is no path connecting vertices x_i and y_i in G.
... | LiveCodeBench | 1,010 | 1 | |
You are in a store to buy N items. The regular price of the i-th item is P_i yen (the currency in Japan).
You have M coupons. You can use the i-th coupon to buy an item whose regular price is at least L_i yen at a D_i-yen discount.
Here, each coupon can be used only once. Besides, multiple coupons cannot be used for... | LiveCodeBench | 1,031 | 2 | |
There are N rectangular cuboids in a three-dimensional space.
These cuboids do not overlap. Formally, for any two different cuboids among them, their intersection has a volume of 0.
The diagonal of the i-th cuboid is a segment that connects two points (X_{i,1},Y_{i,1},Z_{i,1}) and (X_{i,2},Y_{i,2},Z_{i,2}), and its ed... | LiveCodeBench | 1,052 | 3 | |
You are given a positive integer N. Print a string of length (N+1), s_0s_1\ldots s_N, defined as follows.
For each i = 0, 1, 2, \ldots, N,
- if there is a divisor j of N that is between 1 and 9, inclusive, and i is a multiple of N/j, then s_i is the digit corresponding to the smallest such j (s_i will thus be one of ... | LiveCodeBench | 1,074 | 1 | |
The programming contest World Tour Finals is underway, where N players are participating, and half of the competition time has passed.
There are M problems in this contest, and the score A_i of problem i is a multiple of 100 between 500 and 2500, inclusive.
For each i = 1, \ldots, N, you are given a string S_i that ind... | LiveCodeBench | 1,095 | 2 | |
You are given an integer B.
If there exists a positive integer A such that A^A = B, print its value; otherwise, output -1.
Input
The input is given from Standard Input in the following format:
B
Output
If there exists a positive integer A such that A^A = B, print its value; otherwise, print -1.
If there are multipl... | LiveCodeBench | 1,116 | 3 | |
You are given a sequence A=(A_1,\ldots,A_N) of length N.
For each i=1,\ldots,N, solve the following problem.
Problem: Find the sum of all elements in A that are greater than A_i.
Input
The input is given from Standard Input in the following format:
N
A_1 \ldots A_N
Output
For each 1\leq k\leq N, let B_k be the ans... | LiveCodeBench | 1,138 | 1 | |
For a positive integer X, the Dragon String of level X is a string of length (X+3) formed by one L, X occurrences of o, one n, and one g arranged in this order.
You are given a positive integer N. Print the Dragon String of level N.
Note that uppercase and lowercase letters are distinguished.
Input
The input is give... | LiveCodeBench | 1,159 | 2 | |
There is a single integer N written on a blackboard.
Takahashi will repeat the following series of operations until all integers not less than 2 are removed from the blackboard:
- Choose one integer x not less than 2 written on the blackboard.
- Erase one occurrence of x from the blackboard. Then, write two new integ... | LiveCodeBench | 1,180 | 3 | |
You are given N integers A_1,A_2,\dots,A_N, one per line, over N lines. However, N is not given in the input.
Furthermore, the following is guaranteed:
- A_i \neq 0 ( 1 \le i \le N-1 )
- A_N = 0
Print A_N, A_{N-1},\dots,A_1 in this order.
Input
The input is given from Standard Input in the following format:
A_1
A... | LiveCodeBench | 1,202 | 1 | |
For non-negative integers l and r (l < r), let S(l, r) denote the sequence (l, l+1, \ldots, r-2, r-1) formed by arranging integers from l through r-1 in order. Furthermore, a sequence is called a good sequence if and only if it can be represented as S(2^i j, 2^i (j+1)) using non-negative integers i and j.
You are given... | LiveCodeBench | 1,223 | 2 | |
For strings x and y, define f(x, y) as follows:
- f(x, y) is the length of the longest common prefix of x and y.
You are given N strings (S_1, \ldots, S_N) consisting of lowercase English letters. Find the value of the following expression:
\displaystyle \sum_{i=1}^{N-1}\sum_{j=i+1}^N f(S_i,S_j).
Input
The input is... | LiveCodeBench | 1,244 | 3 | |
The problem involves tracking the frequency of IDs in a collection that changes over time. You have two integer arrays, nums and freq, of equal length n. Each element in nums represents an ID, and the corresponding element in freq indicates how many times that ID should be added to or removed from the collection at eac... | LiveCodeBench | 1,266 | 1 | |
You are given a string word. A letter is called special if it appears both in lowercase and uppercase in word.
Return the number of special letters in word.
Example 1:
Input: word = "aaAbcBC"
Output: 3
Explanation:
The special characters in word are 'a', 'b', and 'c'.
Example 2:
Input: word = "abc"
Output: 0
Expla... | LiveCodeBench | 1,287 | 2 | |
You are given an array nums, where each number in the array appears either once or twice.
Return the bitwise XOR of all the numbers that appear twice in the array, or 0 if no number appears twice.
Example 1:
Input: nums = [1,2,1,3]
Output: 1
Explanation:
The only number that appears twice in nums is 1.
Example 2:
... | LiveCodeBench | 1,308 | 3 |
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