problem stringlengths 37 3.95k | answer stringlengths 0 216 | task stringclasses 6
values | problem_number int64 13 1.3k | set_number int64 1 5 |
|---|---|---|---|---|
Find the largest possible real part of \[(75+117i)z+\frac{96+144i}{z}\]where $z$ is a complex number with $|z|=4$. | 540 | AIME24 | 13 | 3 |
Let ABCDEF be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AB, CD, and EF has side lengths 200, 240, and 300. Find the side length of the hexagon. | 080 | AIME24 | 26 | 2 |
The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime.... | 62 | AIME25 | 39 | 1 |
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such tha... | 293 | AIME25 | 51 | 5 |
What is the value of
\[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\] | 3159 | AMC23 | 64 | 4 |
How many positive perfect squares less than $2023$ are divisible by $5$? | 8 | AMC23 | 77 | 3 |
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ Find the area of $ABC$. The final answer can be simplified in the form $m \sqrt{n}$, where $m$ and $n$ are positive integers and $n$ without square factore. What is $m+n$? | 18 | AMC23 | 90 | 2 |
All the following statements about the molecular biology of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS‑CoV‑2) are correct except
| SARS-CoV-2 nsp10/nsp14-ExoN operates as heterodimers in a mismatch repair mechanism. The N-terminal ExoN domain of nsp14 could bind to nsp10 making an active exonuclease complex that prevents the breakdown of dsRNA. | GPQADiamond | 111 | 1 |
You want to study the role of the GADD45G protein in a mouse model. You are highly interested in whether the IL-2-induced protein has a role in the SARS-CoV-2 infection. You decide to create a transgenic mouse model with the inducible overexpression of the coding sequence for the GADD45G gene. You can use it to affinit... | The ribosome terminated the translation early | GPQADiamond | 175 | 1 |
Two compounds A and B are treated separately with hydrochloric acid, sodium nitrite, and water. Through the mentioned treatment two different diketones are produced. Select the proper starting materials A and B for the reactions.
A + (NaNO2, HCl, H2O) ---> 4-isopropylcyclohexane-1,2-dione
B + (NaNO2, HCl, H2O) ---> 5-m... | A = 4-isopropylcyclohexan-1-one, B = 5-methylhexan-2-one | GPQADiamond | 239 | 1 |
The results of a cross-country team's training run are graphed below. Which student has the greatest average speed? [asy]
for ( int i = 1; i <= 7; ++i )
{
draw((i,0)--(i,6));
}
for ( int i = 1; i <= 5; ++i )
{
draw((0,i)--(8,i));
}
draw((-0.5,0)--(8,0), linewidth(1));
draw((0,-0.5)--(0,6), linewidth(1));
label("$O$"... | \text{Evelyn} | MATH500 | 303 | 1 |
For some value of $x,$ $0 < x < 180,$
\[\tan 53^\circ \tan 81^\circ \tan x^\circ = \tan 53^\circ + \tan 81^\circ + \tan x^\circ.\]Find $x.$ | 46 | MATH500 | 367 | 1 |
Rick is thinking of a positive factor of $14$ and Steve is thinking of a positive factor of $42$. If Rick and Steve are thinking of the same number, how many possible numbers could they be thinking of? | 4 | MATH500 | 431 | 1 |
$441+2(21)(19)+361=x$. Solve for $x$. | 1600 | MATH500 | 495 | 1 |
Let $A = (1,-11,2),$ $B = (3,-4,1),$ and $C = (-2,1,-1).$ Compute $\angle ABC,$ in degrees. | 120^\circ | MATH500 | 559 | 1 |
In the diagram, two circles, each with center $D$, have radii of $1$ and $2$. The total area of the shaded region is $\frac5{12}$ of the area of the larger circle. How many degrees are in the measure of (the smaller) $\angle ADC$?
[asy]
size(4cm);
defaultpen(linewidth(0.75));
real adc = 100;
pair d = (0, 0); pair a = 2... | 120 | MATH500 | 623 | 1 |
What is the smallest real number $x$ in the domain of the function $$g(x) = \sqrt{(x-3)^2-(x-8)^2}~?$$ | \frac{11}{2} | MATH500 | 687 | 1 |
The sum of the digits of a two-digit number is $13.$ The difference between the number and the number with its digits reversed is $27.$ What is the sum of the original number and the number with its digits reversed? | 143 | MATH500 | 751 | 1 |
Vanya and Vova are playing a game. Players are given an integer $n$. On their turn, the player can add $1$ to the current integer or subtract $1$. The players take turns; Vanya starts. If after Vanya's move the integer is divisible by $3$, then he wins. If $10$ moves have passed and Vanya has not won, then Vova wins.
... | LiveCodeBench | 804 | 2 | |
You are given a 0-indexed integer array nums of size n representing the cost of collecting different chocolates. The cost of collecting the chocolate at the index i is nums[i]. Each chocolate is of a different type, and initially, the chocolate at the index i is of i^th type.
In one operation, you can do the following ... | LiveCodeBench | 825 | 3 | |
You are given a 0-indexed array words containing n strings.
Let's define a join operation join(x, y) between two strings x and y as concatenating them into xy. However, if the last character of x is equal to the first character of y, one of them is deleted.
For example join("ab", "ba") = "aba" and join("ab", "cde") = "... | LiveCodeBench | 847 | 1 | |
An element x of an integer array arr of length m is dominant if freq(x) * 2 > m, where freq(x) is the number of occurrences of x in arr. Note that this definition implies that arr can have at most one dominant element.
You are given a 0-indexed integer array nums of length n with one dominant element.
You can split num... | LiveCodeBench | 868 | 2 | |
You are given two 0-indexed strings str1 and str2.
In an operation, you select a set of indices in str1, and for each index i in the set, increment str1[i] to the next character cyclically. That is 'a' becomes 'b', 'b' becomes 'c', and so on, and 'z' becomes 'a'.
Return true if it is possible to make str2 a subsequence... | LiveCodeBench | 889 | 3 | |
You are given a binary string s and a positive integer k.
A substring of s is beautiful if the number of 1's in it is exactly k.
Let len be the length of the shortest beautiful substring.
Return the lexicographically smallest beautiful substring of string s with length equal to len. If s doesn't contain a beautiful sub... | LiveCodeBench | 911 | 1 | |
You are given an integer n.
A string s is called good if it contains only lowercase English characters and it is possible to rearrange the characters of s such that the new string contains "leet" as a substring.
For example:
The string "lteer" is good because we can rearrange it to form "leetr" .
"letl" is not good be... | LiveCodeBench | 932 | 2 | |
You are given a 0-indexed string s and an integer k.
You are to perform the following partitioning operations until s is empty:
Choose the longest prefix of s containing at most k distinct characters.
Delete the prefix from s and increase the number of partitions by one. The remaining characters (if any) in s maintain... | LiveCodeBench | 953 | 3 | |
You are given two 1-indexed integer arrays, nums and, changeIndices, having lengths n and m, respectively.
Initially, all indices in nums are unmarked. Your task is to mark all indices in nums.
In each second, s, in order from 1 to m (inclusive), you can perform one of the following operations:
Choose an index i in th... | LiveCodeBench | 975 | 1 | |
There is a grid with H horizontal rows and W vertical columns. Each cell has a lowercase English letter written on it.
We denote by (i, j) the cell at the i-th row from the top and j-th column from the left.
The letters written on the grid are represented by H strings S_1,S_2,\ldots, S_H, each of length W.
The j-th ... | LiveCodeBench | 996 | 2 | |
You are given a sequence A=(A_0,A_1,\dots,A_{63}) of length 64 consisting of 0 and 1.
Find A_0 2^0 + A_1 2^1 + \dots + A_{63} 2^{63}.
Input
The input is given from Standard Input in the following format:
A_0 A_1 \dots A_{63}
Output
Print the answer as an integer.
Constraints
- A_i is 0 or 1.
Sample Input 1
1 0... | LiveCodeBench | 1,017 | 3 | |
There are N sticks with several balls stuck onto them. Each ball has a lowercase English letter written on it.
For each i = 1, 2, \ldots, N, the letters written on the balls stuck onto the i-th stick are represented by a string S_i.
Specifically, the number of balls stuck onto the i-th stick is the length |S_i| of the... | LiveCodeBench | 1,039 | 1 | |
You are given a string S of length N consisting of uppercase and lowercase English letters.
Let us perform Q operations on the string S.
The i-th operation (1\leq i\leq Q) is represented by a tuple (t _ i,x _ i,c _ i) of two integers and one character, as follows.
- If t _ i=1, change the x _ i-th character of S to c ... | LiveCodeBench | 1,060 | 2 | |
There are N people numbered 1 to N on a coordinate plane.
Person 1 is at the origin.
You are given M pieces of information in the following form:
- From person A_i's perspective, person B_i is X_i units away in the positive x-direction and Y_i units away in the positive y-direction.
Determine the coordinates of each... | LiveCodeBench | 1,081 | 3 | |
There is a directed graph with N vertices and M edges. Each edge has two positive integer values: beauty and cost.
For i = 1, 2, \ldots, M, the i-th edge is directed from vertex u_i to vertex v_i, with beauty b_i and cost c_i.
Here, the constraints guarantee that u_i \lt v_i.
Find the maximum value of the following fo... | LiveCodeBench | 1,103 | 1 | |
You are given a weighted simple connected undirected graph with N vertices and M edges, where vertices are numbered 1 to N, and edges are numbered 1 to M. Additionally, a positive integer K is given.
Edge i\ (1\leq i\leq M) connects vertices u_i and v_i and has a weight of w_i.
For a spanning tree T of this graph, the ... | LiveCodeBench | 1,124 | 2 | |
You are given an integer N between 1 and 9, inclusive, as input.
Concatenate N copies of the digit N and print the resulting string.
Input
The input is given from Standard Input in the following format:
N
Output
Print the answer.
Constraints
- N is an integer between 1 and 9, inclusive.
Sample Input 1
3
Sampl... | LiveCodeBench | 1,145 | 3 | |
This is an interactive problem (a type of problem where your program interacts with the judge program through Standard Input and Output).
There are N bottles of juice, numbered 1 to N. It has been discovered that exactly one of these bottles has gone bad. Even a small sip of the spoiled juice will cause stomach upset t... | LiveCodeBench | 1,167 | 1 | |
You are given a simple undirected graph consisting of N vertices and M edges.
For i = 1, 2, \ldots, M, the i-th edge connects vertices u_i and v_i.
Also, for i = 1, 2, \ldots, N, vertex i is assigned a positive integer W_i, and there are A_i pieces placed on it.
As long as there are pieces on the graph, repeat the foll... | LiveCodeBench | 1,188 | 2 | |
You are given N integers A_1, A_2, \dots, A_N.
Also, define B_i = A_i \times A_{i+1}\ (1 \leq i \leq N-1).
Print B_1, B_2, \dots, B_{N-1} in this order, separated by spaces.
Input
The input is given from Standard Input in the following format:
N
A_1 A_2 \dots A_N
Output
Print B_1, B_2, \dots, B_{N-1} in this orde... | LiveCodeBench | 1,209 | 3 | |
You are given two grids, each with N rows and N columns, referred to as grid A and grid B.
Each cell in the grids contains a lowercase English letter.
The character at the i-th row and j-th column of grid A is A_{i, j}.
The character at the i-th row and j-th column of grid B is B_{i, j}.
The two grids differ in ex... | LiveCodeBench | 1,231 | 1 | |
You are given a sequence A=(A_1,A_2,\dots,A_N) of length N and a sequence B=(B_1,B_2,\dots,B_M) of length M. Here, all elements of A and B are pairwise distinct. Determine whether the sequence C=(C_1,C_2,\dots,C_{N+M}) formed by sorting all elements of A and B in ascending order contains two consecutive elements appear... | LiveCodeBench | 1,252 | 2 | |
You are given an integer array nums and a non-negative integer k. In one operation, you can increase or decrease any element by 1.
Return the minimum number of operations needed to make the median of nums equal to k.
The median of an array is defined as the middle element of the array when it is sorted in non-decreasin... | LiveCodeBench | 1,273 | 3 | |
You are given a 2D matrix grid of size 3 x 3 consisting only of characters 'B' and 'W'. Character 'W' represents the white color, and character 'B' represents the black color.
Your task is to change the color of at most one cell so that the matrix has a 2 x 2 square where all cells are of the same color.
Return true if... | LiveCodeBench | 1,295 | 1 |
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