problem stringlengths 46 3.88k | answer stringlengths 0 118 | 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 | 3 |
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 | 2 |
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 | 1 |
A piecewise linear function is defined by\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(... | 259 | AIME25 | 41 | 5 |
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 | 4 |
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 | 3 |
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 | 2 |
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 | 1 |
Name reactions in chemistry refer to a specific set of well-known chemical reactions that are typically named after their discoverers or the scientists who made significant contributions to their development. These reactions have had a profound impact on the field of chemistry and are often used as fundamental building... | A = 2,7-dimethyloctahydronaphthalene-4a,8a-diol, B = (((3-methylbut-2-en-1-yl)oxy)methyl)benzene | GPQADiamond | 125 | 1 |
1,3-dibromoadamantane is heated to 240C with an excess of KOH, creating a light yellow solid product 1, With 1H nmr spectrum: 4.79(2H), 2.41-2.23(10H), 1.94(2H) and a characteristic absorbance in the IR spectrum at 1720cm-1.
This product was then heated with an excess of aluminum isopropoxide, forming product 2.
ozon... | triplet of triplets | GPQADiamond | 189 | 1 |
α-β unsaturated carbonyls have a much more electrophilic carbon double bond. These double bonds act as a good Lewis acid and react with nucleophiles to produce enolate ions. In this case, when a nucleophile attacks the β position, it produces a resonance-stabilized intermediate. When a nucleophile attacking the β-carbo... | A = trimethyl 2-(p-tolyl)propane-1,1,3-tricarboxylate, B = 3-(2-oxocyclohexyl)butanenitrile, C = cyclohexane-1,3-dione | GPQADiamond | 253 | 1 |
$\overline{BC}$ is parallel to the segment through $A$, and $AB = BC$. What is the number of degrees represented by $x$?
[asy]
draw((0,0)--(10,0));
draw((0,3)--(10,3));
draw((2,3)--(8,0));
draw((2,3)--(4,0));
label("$A$",(2,3),N);
label("$B$",(4,0),S);
label("$C$",(8,0),S);
label("$124^{\circ}$",(2,3),SW);
label("$x^{... | 28 | MATH500 | 317 | 1 |
Point $A$ lies somewhere within or on the square which has opposite corners at $(0,0)$ and $(2,2)$. Point $B$ lies somewhere within or on the square which has opposite corners at points $(4,2)$ and $(5,3)$. What is the greatest possible value of the slope of the line containing points $A$ and $B$? Express your answer a... | \frac{3}{2} | MATH500 | 381 | 1 |
Simplify the following:$$\frac{3}{\sqrt{27}}$$ | \frac{\sqrt{3}}{3} | MATH500 | 445 | 1 |
Twelve 1 by 1 squares form a rectangle, as shown. What is the total area of the shaded region? [asy]
size(4cm);defaultpen(linewidth(0.75));
// Filled portions
fill((0, 4)--(0, 0)--(2, 0)--cycle, gray(0.75));
fill((0, 4)--(3, 4)--(3, 0)--cycle, gray(0.75));
// grid
int j;
for (j = 0; j < 4; ++j) {draw((j, 0)--(j, 4));... | 10 | MATH500 | 509 | 1 |
Let $S$ be the union of the set of all points inside a regular nonagon with side length $2$ units and the set of all points less than $1$ unit away from a point on the perimeter of the nonagon. What, in units, is the perimeter of $S$? | 18+2\pi | MATH500 | 573 | 1 |
What is the base six equivalent of $999_{10}$? | 4343_6 | MATH500 | 637 | 1 |
A $90^\circ$ rotation around $-4 - 5i$ in the clockwise direction is applied to $3$ (as a complex number). What is the resulting complex number? | 1 - 12i | MATH500 | 701 | 1 |
If $re^{i \theta}$ is a root of
\[z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1 = 0,\]where $r > 0$ and $0 \le \theta < 2 \pi,$ then find the sum of all possible values of $\theta.$ | 8 \pi | MATH500 | 765 | 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 | 1 | |
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 | 2 | |
You are given a 0-indexed integer array nums and an integer threshold.
Find the length of the longest subarray of nums starting at index l and ending at index r (0 <= l <= r < nums.length) that satisfies the following conditions:
nums[l] % 2 == 0
For all indices i in the range [l, r - 1], nums[i] % 2 != nums[i + 1] % ... | LiveCodeBench | 851 | 3 | |
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 | 1 | |
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 | 2 | |
You are given a 0-indexed integer array nums.
The distinct count of a subarray of nums is defined as:
Let nums[i..j] be a subarray of nums consisting of all the indices from i to j such that 0 <= i <= j < nums.length. Then the number of distinct values in nums[i..j] is called the distinct count of nums[i..j].
Return ... | LiveCodeBench | 915 | 3 | |
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 | 1 | |
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 | 2 | |
You are given a string s.
Consider performing the following operation until s becomes empty:
For every alphabet character from 'a' to 'z', remove the first occurrence of that character in s (if it exists).
For example, let initially s = "aabcbbca". We do the following operations:
Remove the underlined characters s =... | LiveCodeBench | 979 | 3 | |
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 | 1 | |
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 | 2 | |
You are given a string S consisting of A, B, and C. S is guaranteed to contain all of A, B, and C.
If the characters of S are checked one by one from the left, how many characters will have been checked when the following condition is satisfied for the first time?
- All of A, B, and C have appeared at least once.
Inp... | LiveCodeBench | 1,043 | 3 | |
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 | 1 | |
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 | 2 | |
There are N products labeled 1 to N flowing on a conveyor belt.
A Keyence printer is attached to the conveyor belt, and product i enters the range of the printer T_i microseconds from now and leaves it D_i microseconds later.
The Keyence printer can instantly print on one product within the range of the printer (in pa... | LiveCodeBench | 1,107 | 3 | |
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 | 1 | |
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 | 2 | |
The AtCoder Archipelago consists of N islands connected by N bridges.
The islands are numbered from 1 to N, and the i-th bridge (1\leq i\leq N-1) connects islands i and i+1 bidirectionally, while the N-th bridge connects islands N and 1 bidirectionally.
There is no way to travel between islands other than crossing the ... | LiveCodeBench | 1,171 | 3 | |
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 | 1 | |
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 | 2 | |
There is a grid of H rows and W columns. Some cells (possibly zero) contain magnets.
The state of the grid is represented by H strings S_1, S_2, \ldots, S_H of length W. If the j-th character of S_i is #, it indicates that there is a magnet in the cell at the i-th row from the top and j-th column from the left; if it i... | LiveCodeBench | 1,235 | 3 | |
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 | 1 | |
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 | 2 | |
In a mystic dungeon, n magicians are standing in a line. Each magician has an attribute that gives you energy. Some magicians can give you negative energy, which means taking energy from you.
You have been cursed in such a way that after absorbing energy from magician i, you will be instantly transported to magician (i... | LiveCodeBench | 1,299 | 3 |
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