problem stringlengths 34 3.88k | answer stringlengths 0 41 | task stringclasses 6
values | problem_number int64 3 1.3k | set_number int64 1 5 |
|---|---|---|---|---|
Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integ... | 371 | AIME24 | 3 | 4 |
Among the 900 residents of Aimeville, there are 195 who own a diamond ring, 367 who own a set of golf clubs, and 562 who own a garden spade. In addition, each of the 900 residents owns a bag of candy hearts. There are 437 residents who own exactly two of these things, and 234 residents who own exactly three of these th... | 073 | AIME24 | 16 | 3 |
Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$. When $T$ rests on the outside of $S$, it is externally tangent to $S$ along a circle with radi... | 127 | AIME24 | 29 | 2 |
The set of points in $3$-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities\[x-yz<y-zx<z-xy\]forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$... | 510 | AIME25 | 42 | 1 |
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$. | 149 | AIME25 | 54 | 5 |
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 21 | AMC23 | 67 | 4 |
Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$? The final answer can be written in the form $m \sqrt{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 9 | AMC23 | 80 | 3 |
Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.
\[abcd=2^6\cdot 3^9\cdot 5^7\]
\[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\]
\[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\]
\[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\]
\[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\]
\[\text{lcm}(b,d)=2^2... | 3 | AMC23 | 93 | 2 |
Dienes are organic compounds with two adjacent double bonds in their structure, and they exhibit unique reactivity due to their conjugated pi-electron system. They play a significant role in organic chemistry and are involved in various chemical reactions and natural processes.
Among the given options which one is the ... | A = 2,2-diiodoethen-1-one, B = 3, 1, 2, 4 | GPQADiamond | 126 | 1 |
Determine the product obtained when 3,4-dimethylhexanedial is reacted with the following set of reagents:
1. KOH, H2O, THF, Heat
2. CH3CH2MgBr, H3O+
3. PCC, CH2Cl2
4. O3, H2O | 3,4-dimethyl-5,6-dioxooctanoic acid | GPQADiamond | 190 | 1 |
In the lab, a chemist discovers an unnamed bottle with a white solid inside. He gathers the following spectral information in order to identify the liquid. Make a reasonable structural suggestion for the unidentified drug using the information provided.
Mass: molecular ion peak at m/z = 156 (100%)
a peak at m/z = 158 (... | 4-chlorobenzoic acid | GPQADiamond | 254 | 1 |
Let $a$ be a positive real number such that all the roots of
\[x^3 + ax^2 + ax + 1 = 0\]are real. Find the smallest possible value of $a.$ | 3 | MATH500 | 318 | 1 |
Solve for $x: 3^{2x} + 19 = 10^x$. | 2 | MATH500 | 382 | 1 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 2.$ Find the minimum value of
\[x^4 + 4y^2 + 4z^4.\] | 16 | MATH500 | 446 | 1 |
Express $5^5\div5^4-5^3+5^2\cdot5^1$ as an integer. | 5 | MATH500 | 510 | 1 |
Let $f(x) = 2x-3$ and $g(x) = x+1$. What is the value of $g(f(5)-1)$? | 7 | MATH500 | 574 | 1 |
The parabola with equation $y=ax^2+bx+c$ and vertex $(h,k)$ is reflected about the line $y=k$. This results in the parabola with equation $y=dx^2+ex+f$. Express $a+b+c+d+e+f$ in terms of $k.$ | 2k | MATH500 | 638 | 1 |
How many of the same digits are found in the base 7 and base 8 representations of $629_{10}$? For example, $121_{3}$ and $413_{5}$ would have one digit in common. | 3 | MATH500 | 702 | 1 |
In the diagram, what is the area of the shaded triangle? [asy]
fill((0,0)--(2,3)--(10,0)--cycle,gray);
draw((0,0)--(10,0)--(10,3)--(0,3)--cycle,linewidth(1));
draw((0,0)--(2,3)--(10,0),linewidth(1));
label("10 cm",(5,3),N);
label("10 cm",(5,0),S);
label("3 cm",(0,1.5),W);
label("3 cm",(10,1.5),E);
draw((0,2.5)--(.5,2.5... | 15\mbox{ cm}^2 | MATH500 | 766 | 1 |
You are given a 0-indexed 2D integer array nums. Initially, your score is 0. Perform the following operations until the matrix becomes empty:
From each row in the matrix, select the largest number and remove it. In the case of a tie, it does not matter which number is chosen.
Identify the highest number amongst all th... | LiveCodeBench | 809 | 2 | |
Given a positive integer num represented as a string, return the integer num without trailing zeros as a string.
Example 1:
Input: num = "51230100"
Output: "512301"
Explanation: Integer "51230100" has 2 trailing zeros, we remove them and return integer "512301".
Example 2:
Input: num = "123"
Output: "123"
Explanat... | LiveCodeBench | 830 | 3 | |
You are given a binary array nums.
A subarray of an array is good if it contains exactly one element with the value 1.
Return an integer denoting the number of ways to split the array nums into good subarrays. As the number may be too large, return it modulo 10^9 + 7.
A subarray is a contiguous non-empty sequence of el... | LiveCodeBench | 852 | 1 | |
You are given a 0-indexed integer array nums, an integer modulo, and an integer k.
Your task is to find the count of subarrays that are interesting.
A subarray nums[l..r] is interesting if the following condition holds:
Let cnt be the number of indices i in the range [l, r] such that nums[i] % modulo == k. Then, cnt %... | LiveCodeBench | 873 | 2 | |
You are given a 0-indexed integer array receiver of length n and an integer k.
There are n players having a unique id in the range [0, n - 1] who will play a ball passing game, and receiver[i] is the id of the player who receives passes from the player with id i. Players can pass to themselves, i.e. receiver[i] may be ... | LiveCodeBench | 894 | 3 | |
Given a 0-indexed array of strings words where words[i] is either a positive integer represented as a string or the string "prev".
Start iterating from the beginning of the array; for every "prev" string seen in words, find the last visited integer in words which is defined as follows:
Let k be the number of consecuti... | LiveCodeBench | 916 | 1 | |
You are given a string s and a positive integer k.
Let vowels and consonants be the number of vowels and consonants in a string.
A string is beautiful if:
vowels == consonants.
(vowels * consonants) % k == 0, in other terms the multiplication of vowels and consonants is divisible by k.
Return the number of non-empty ... | LiveCodeBench | 937 | 2 | |
You are given an integer k and an integer x.
Consider s is the 1-indexed binary representation of an integer num. The price of a number num is the number of i's such that i % x == 0 and s[i] is a set bit.
Return the greatest integer num such that the sum of prices of all numbers from 1 to num is less than or equal to k... | LiveCodeBench | 958 | 3 | |
You are given a 0-indexed string array words.
Let's define a boolean function isPrefixAndSuffix that takes two strings, str1 and str2:
isPrefixAndSuffix(str1, str2) returns true if str1 is both a prefix and a suffix of str2, and false otherwise.
For example, isPrefixAndSuffix("aba", "ababa") is true because "aba" is ... | LiveCodeBench | 980 | 1 | |
Two characters x and y are called similar characters if and only if one of the following conditions is satisfied:
- x and y are the same character.
- One of x and y is 1 and the other is l.
- One of x and y is 0 and the other is o.
Two strings S and T, each of length N, are called similar strings if and only if:
- f... | LiveCodeBench | 1,001 | 2 | |
You are given N strings S_1,S_2,\ldots,S_N consisting of lowercase English letters.
Determine if there are distinct integers i and j between 1 and N, inclusive, such that the concatenation of S_i and S_j in this order is a palindrome.
A string T of length M is a palindrome if and only if the i-th character and the (M+... | LiveCodeBench | 1,022 | 3 | |
There are N people numbered 1 to N.
You are given their schedule for the following D days. The schedule for person i is represented by a string S_i of length D. If the j-th character of S_i is o, person i is free on the j-th day; if it is x, they are occupied that day.
From these D days, consider choosing some consecu... | LiveCodeBench | 1,044 | 1 | |
We have N cups of ice cream.
The flavor and deliciousness of the i-th cup are F_i and S_i, respectively (S_i is an even number).
You will choose and eat two of the N cups.
Your satisfaction here is defined as follows.
- Let s and t (s \ge t) be the deliciousness of the eaten cups.
- If the two cups have different ... | LiveCodeBench | 1,065 | 2 | |
AtCoder cafeteria offers N main dishes and M side dishes. The price of the i-th main dish is A_i, and that of the j-th side dish is B_j.
The cafeteria is considering introducing a new set meal menu.
A set meal consists of one main dish and one side dish. Let s be the sum of the prices of the main dish and the side di... | LiveCodeBench | 1,086 | 3 | |
There are N cities in a certain country.
You will travel from your office in city 1 to a destination in city N, via zero or more cities.
Two types of transportation are available: company car and train. The time required to travel from city i to city j is as follows:
- D_{i,j} \times A minutes by company car, and
- D_... | LiveCodeBench | 1,108 | 1 | |
You are given two strings: S, which consists of uppercase English letters and has length N, and T, which also consists of uppercase English letters and has length M\ (\leq N).
There is a string X of length N consisting only of the character #. Determine whether it is possible to make X match S by performing the followi... | LiveCodeBench | 1,129 | 2 | |
Takahashi, a young baseball enthusiast, has been a very good boy this year, so Santa has decided to give him a bat or a glove, whichever is more expensive.
If a bat costs B yen and a glove costs G yen (B\neq G), which one will Santa give to Takahashi?
Input
The input is given from Standard Input in the following form... | LiveCodeBench | 1,150 | 3 | |
There are 2N points placed at equal intervals on a circle, numbered 1 to 2N in a clockwise direction starting from a certain point.
There are also N chords on the circle, with the i-th chord connecting points A_i and B_i.
It is guaranteed that all the values A_1,\dots,A_N,B_1,\dots,B_N are distinct.
Determine whether t... | LiveCodeBench | 1,172 | 1 | |
In the country of AtCoder, there are N stations: station 1, station 2, \ldots, station N.
You are given M pieces of information about trains in the country. The i-th piece of information (1\leq i\leq M) is represented by a tuple of six positive integers (l _ i,d _ i,k _ i,c _ i,A _ i,B _ i), which corresponds to the fo... | LiveCodeBench | 1,193 | 2 | |
You are given a string S consisting of lowercase English letters. How many different non-empty substrings does S have?
A substring is a contiguous subsequence. For example, xxx is a substring of yxxxy but not of xxyxx.
Input
The input is given from Standard Input in the following format:
S
Output
Print the answer.
... | LiveCodeBench | 1,214 | 3 | |
You are given a weighted undirected graph G with N vertices, numbered 1 to N. Initially, G has no edges.
You will perform M operations to add edges to G. The i-th operation (1 \leq i \leq M) is as follows:
- You are given a subset of vertices S_i=\lbrace A_{i,1},A_{i,2},\dots,A_{i,K_i}\rbrace consisting of K_i vertice... | LiveCodeBench | 1,236 | 1 | |
You are given an array happiness of length n, and a positive integer k.
There are n children standing in a queue, where the i^th child has happiness value happiness[i]. You want to select k children from these n children in k turns.
In each turn, when you select a child, the happiness value of all the children that hav... | LiveCodeBench | 1,257 | 2 | |
You are given an integer array nums containing positive integers. We define a function encrypt such that encrypt(x) replaces every digit in x with the largest digit in x. For example, encrypt(523) = 555 and encrypt(213) = 333.
Return the sum of encrypted elements.
Example 1:
Input: nums = [1,2,3]
Output: 6
Explanati... | LiveCodeBench | 1,278 | 3 | |
An array is considered special if every pair of its adjacent elements contains two numbers with different parity.
You are given an array of integers nums. Return true if nums is a special array, otherwise, return false.
Example 1:
Input: nums = [1]
Output: true
Explanation:
There is only one element. So the answer i... | LiveCodeBench | 1,300 | 1 |
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