problem stringlengths 43 2.71k | answer stringlengths 0 53 | task stringclasses 6
values | problem_number int64 11 1.29k | set_number int64 1 5 |
|---|---|---|---|---|
Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16$,$AB=107$,$FG=17$, and $EF=184$, what is the length of $CE$? | 104 | AIME24 | 11 | 5 |
Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is $99$. The grid below is an example of such an arrangement because $8+991=999$ and $9+9+81=99$.
\[\begin{arr... | 045 | AIME24 | 24 | 4 |
The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB... | 821 | AIME25 | 37 | 3 |
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $ riangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$... | 336 | AIME25 | 50 | 2 |
What is the degree measure of the acute angle formed by lines with slopes $2$ and $\frac{1}{3}$? | 45 | AMC23 | 63 | 1 |
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}... | 5 | AMC23 | 75 | 5 |
For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots? | 5 | AMC23 | 88 | 4 |
A spin-half particle is in a linear superposition 0.5|\uparrow\rangle+sqrt(3)/2|\downarrow\rangle of its spin-up and spin-down states. If |\uparrow\rangle and |\downarrow\rangle are the eigenstates of \sigma{z} , then what is the expectation value up to one decimal place, of the operator 10\sigma{z}+5\sigma_{x} ? Here,... | -0.7 | GPQADiamond | 103 | 1 |
Consider two electrons are in p orbital angular momentum state. Ignoring there spin angular momentum, the coupled angular momentum state of the 2-electron system is represented by $|l_1, l_2, l, m> = |1,1, 2, -1>. Here, $l_1, l_2$ are the orbital angular momentum quantum numbers of particles 1 and 2 respectively. And, ... | 0 | GPQADiamond | 167 | 1 |
A dicarboxylic acid containing a cis-alkene was dehydrated to the corresponding anhydride. This anhydride, which has a single peak (7 ppm) in its 1H NMR spectrum and two peaks (137 ppm, 165 ppm) in its 13C spectrum, was reacted with 1,2,3,4-tetramethyl-1,3-cyclopentadiene to yield two products (a major and a minor prod... | A 1H doublet at ~1.5 ppm and a 2H singlet at ~3.5 ppm | GPQADiamond | 231 | 1 |
Astronomers are observing a planet with a radius of 1 Earth radius in a circular orbit, with a transit impact parameter of 0.2, around a star that has a radius 1.5 times that of the Sun. This planet has an orbital period of 3 days. What is the maximum orbital period of a second planet, with a radius of 2.5 times that o... | ~ 33.5 | GPQADiamond | 295 | 1 |
The polynomial $x^3 - 3x^2 + 4x - 1$ is a factor of $x^9 + px^6 + qx^3 + r.$ Enter the ordered triple $(p,q,r).$ | (6,31,-1) | MATH500 | 359 | 1 |
If $|x+5|-|3x-6|=0$, find the largest possible value of $x$. Express your answer as an improper fraction. | \frac{11}{2} | MATH500 | 423 | 1 |
Find the largest $x$-value at which the graphs of $f(x)=e^{3x^2-|\lfloor x \rfloor|!}+\binom{22+735235|\lfloor x \rfloor |}{2356}+\phi(|\lfloor x \rfloor|+1)+72x^4+3x^3-6x^2+2x+1$ and $g(x)=e^{3x^2-|\lfloor x \rfloor|!}+\binom{22+735235|\lfloor x \rfloor |}{2356}+\phi(|\lfloor x \rfloor|+1)+72x^4+4x^3-11x^2-6x+13$ inte... | 6 | MATH500 | 487 | 1 |
What is the maximum value of $4(x + 7)(2 - x)$, over all real numbers $x$? | 81 | MATH500 | 551 | 1 |
The expression $10x^2-x-24$ can be written as $(Ax-8)(Bx+3),$ where $A$ and $B$ are integers. What is $AB + B$? | 12 | MATH500 | 615 | 1 |
Solve
\[\frac{|x^2 - 81|}{x^2 - 36x} < 0.\] | (0,9) \cup (9,36) | MATH500 | 679 | 1 |
Rectangle $ABCD$ has center $O$ and $AB/AD=k$. A point is randomly chosen from the interior of rectangle $ABCD$. What is the probability that it is closer to $O$ than to any of the four vertices? [asy]
size(200);
draw((-250,100)--(250,100)--(250,-100)--(-250,-100)--cycle);
dot((0,0));
label("$O$",(0,0),N);
label("$A$"... | \frac{1}{2} | MATH500 | 743 | 1 |
You are given a strip of paper $s$ that is $n$ cells long. Each cell is either black or white. In an operation you can take any $k$ consecutive cells and make them all white.
Find the minimum number of operations needed to remove all black cells.
Input
The first line contains a single integer $t$ ($1 \leq t \leq 100... | LiveCodeBench | 801 | 3 | |
Given a positive integer n, return the punishment number of n.
The punishment number of n is defined as the sum of the squares of all integers i such that:
1 <= i <= n
The decimal representation of i * i can be partitioned into contiguous substrings such that the sum of the integer values of these substrings equals i.... | LiveCodeBench | 823 | 1 | |
You are given a 0-indexed integer array nums containing n distinct positive integers. A permutation of nums is called special if:
For all indexes 0 <= i < n - 1, either nums[i] % nums[i+1] == 0 or nums[i+1] % nums[i] == 0.
Return the total number of special permutations. As the answer could be large, return it modulo... | LiveCodeBench | 844 | 2 | |
You are given a string word and an array of strings forbidden.
A string is called valid if none of its substrings are present in forbidden.
Return the length of the longest valid substring of the string word.
A substring is a contiguous sequence of characters in a string, possibly empty.
Example 1:
Input: word = "cb... | LiveCodeBench | 865 | 3 | |
You are given a 0-indexed integer array nums and an integer x.
Find the minimum absolute difference between two elements in the array that are at least x indices apart.
In other words, find two indices i and j such that abs(i - j) >= x and abs(nums[i] - nums[j]) is minimized.
Return an integer denoting the minimum abso... | LiveCodeBench | 887 | 1 | |
You are given a 0-indexed array of integers nums, and an integer target.
Return the length of the longest subsequence of nums that sums up to target. If no such subsequence exists, return -1.
A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of t... | LiveCodeBench | 908 | 2 | |
There are n balls on a table, each ball has a color black or white.
You are given a 0-indexed binary string s of length n, where 1 and 0 represent black and white balls, respectively.
In each step, you can choose two adjacent balls and swap them.
Return the minimum number of steps to group all the black balls to the ri... | LiveCodeBench | 929 | 3 | |
You are given a 0-indexed string word.
In one operation, you can pick any index i of word and change word[i] to any lowercase English letter.
Return the minimum number of operations needed to remove all adjacent almost-equal characters from word.
Two characters a and b are almost-equal if a == b or a and b are adjacent... | LiveCodeBench | 951 | 1 | |
You are given a 0-indexed integer array nums of size n, and a 0-indexed integer array pattern of size m consisting of integers -1, 0, and 1.
A subarray nums[i..j] of size m + 1 is said to match the pattern if the following conditions hold for each element pattern[k]:
nums[i + k + 1] > nums[i + k] if pattern[k] == 1.
n... | LiveCodeBench | 972 | 2 | |
We have a grid with H rows and W columns.
Let (i,j) denote the square at the i-th row from the top and j-th column from the left.
Each square in the grid is one of the following: the start square, the goal square, an empty square, a wall square, and a candy square.
(i,j) is represented by a character A_{i,j}, and is... | LiveCodeBench | 993 | 3 | |
There is a simple undirected graph with N vertices and M edges, where vertices are numbered from 1 to N, and edges are numbered from 1 to M. Edge i connects vertex a_i and vertex b_i.
K security guards numbered from 1 to K are on some vertices. Guard i is on vertex p_i and has a stamina of h_i. All p_i are distinct.
A ... | LiveCodeBench | 1,015 | 1 | |
There is a family consisting of person 1, person 2, \ldots, and person N. For i\geq 2, person i's parent is person p_i.
They bought insurance M times. For i=1,2,\ldots,M, person x_i bought the i-th insurance, which covers that person and their descendants in the next y_i generations.
How many people are covered by ... | LiveCodeBench | 1,036 | 2 | |
The number pi to the 100-th decimal place is
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679.
You are given an integer N between 1 and 100, inclusive.
Print the value of pi to the N-th decimal place.
More precisely, truncate the value of pi to N decimal places and ... | LiveCodeBench | 1,057 | 3 | |
You are given a string S.
Find the maximum length of a contiguous substring of S that is a palindrome.
Note that there is always a contiguous substring of S that is a palindrome.
Input
The input is given from Standard Input in the following format:
S
Output
Print the answer.
Constraints
- S is a string of leng... | LiveCodeBench | 1,079 | 1 | |
Takahashi sent a string T consisting of lowercase English letters to Aoki. As a result, Aoki received a string T' consisting of lowercase English letters.
T' may have been altered from T. Specifically, exactly one of the following four conditions is known to hold.
- T' is equal to T.
- T' is a string obtained by inser... | LiveCodeBench | 1,100 | 2 | |
AtCoder Kingdom uses a calendar whose year has N months.
Month i (1\leq i\leq N) has D _ i days, from day 1 of month i to day D _ i of month i.
How many days in a year of AtCoder have "repdigits" dates?
Here, day j of month i (1\leq i\leq N,1\leq j\leq D _ i) is said to have a repdigit date if and only if all digits in... | LiveCodeBench | 1,121 | 3 | |
AtCoder Inc. sells T-shirts with its logo.
You are given Takahashi's schedule for N days as a string S of length N consisting of 0, 1, and 2.
Specifically, for an integer i satisfying 1\leq i\leq N,
- if the i-th character of S is 0, he has no plan scheduled for the i-th day;
- if the i-th character of S is 1, he plan... | LiveCodeBench | 1,143 | 1 | |
We define Extended A strings, Extended B strings, Extended C strings, and Extended ABC strings as follows:
- A string S is an Extended A string if all characters in S are A.
- A string S is an Extended B string if all characters in S are B.
- A string S is an Extended C string if all characters in S are C.
- A string ... | LiveCodeBench | 1,164 | 2 | |
There is a grid with H rows and W columns.
Each cell of the grid is land or sea, which is represented by H strings S_1, S_2, \ldots, S_H of length W. Let (i, j) denote the cell at the i-th row from the top and j-th column from the left, and (i, j) is land if the j-th character of S_i is ., and (i, j) is sea if the char... | LiveCodeBench | 1,185 | 3 | |
There is an infinitely long piano keyboard.
Is there a continuous segment within this keyboard that consists of W white keys and B black keys?
Let S be the string formed by infinitely repeating the string wbwbwwbwbwbw.
Is there a substring of S that consists of W occurrences of w and B occurrences of b?
What is a sub... | LiveCodeBench | 1,207 | 1 | |
You are given a permutation A=(A_1,\ldots,A_N) of (1,2,\ldots,N).
Transform A into (1,2,\ldots,N) by performing the following operation between 0 and N-1 times, inclusive:
- Operation: Choose any pair of integers (i,j) such that 1\leq i < j \leq N. Swap the elements at the i-th and j-th positions of A.
It can be pro... | LiveCodeBench | 1,228 | 2 | |
Takahashi has N cards from the card game "AtCoder Magics." The i-th card will be called card i. Each card has two parameters: strength and cost. Card i has a strength of A_i and a cost of C_i.
He does not like weak cards, so he will discard them. Specifically, he will repeat the following operation until it can no long... | LiveCodeBench | 1,249 | 3 | |
You are given an array of integers nums. Return the length of the longest subarray of nums which is either strictly increasing or strictly decreasing.
Example 1:
Input: nums = [1,4,3,3,2]
Output: 2
Explanation:
The strictly increasing subarrays of nums are [1], [2], [3], [3], [4], and [1,4].
The strictly decreasing ... | LiveCodeBench | 1,271 | 1 | |
You are given a string word of size n, and an integer k such that k divides n.
In one operation, you can pick any two indices i and j, that are divisible by k, then replace the substring of length k starting at i with the substring of length k starting at j. That is, replace the substring word[i..i + k - 1] with the su... | LiveCodeBench | 1,292 | 2 |
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