problem stringlengths 112 2.56k | answer stringlengths 0 42 | task stringclasses 6
values | problem_number int64 4 1.3k | set_number int64 1 5 |
|---|---|---|---|---|
Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$. Find the number of intersections of the graphs of \[y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).\] | 385 | AIME24 | 4 | 4 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | 468 | AIME24 | 17 | 3 |
There is a collection of $25$ indistinguishable white chips and $25$ indistinguishable black chips. Find the number of ways to place some of these chips in the $25$ unit cells of a $5\times5$ grid such that:
each cell contains at most one chip
all chips in the same row and all chips in the same column have the same c... | 902 | AIME24 | 30 | 2 |
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expect... | 204 | AIME25 | 43 | 1 |
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$. | 907 | AIME25 | 55 | 5 |
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$.
What is the probability that Flora will eventually land at 10? Write the answer as a simplified fraction... | 3 | AMC23 | 68 | 4 |
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-d... | 9 | AMC23 | 81 | 3 |
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. Find the ratio of the area of circle $A$ to the area of circle $B$. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 194 | AMC23 | 94 | 2 |
toluene is treated with nitric acid and sulfuric acid, forming product 1.
1 is treated with MnO2 and H2SO4, forming product 2.
2 is treated with acetone and aqueous sodium hydroxide, forming product 3.
what is the molecular symmetry group of 3? | c2h | GPQADiamond | 131 | 1 |
3,3,6-trimethylhepta-1,5-dien-4-one is treated with 1 equivalent of meta-chloroperbenzoic acid. Two different products are formed, in approximately a 1:1 ratio.
In a separate vessel, Methyllithium is treated with copper(I)iodide. an excess of the resulting solution is slowly added to the product mixture of the first r... | 6-hydroxy-2,2,5,5-tetramethyloctan-4-one | GPQADiamond | 195 | 1 |
Observations of a quasar across the electromagnetic spectrum are being carried out. Focusing on the near infrared and optical spectrum, there is a peak at wavelength of about 790 nm, and at shorter wavelengths < 790 nm the flux drops significantly.
If we lived in a universe defined by Lambda-CDM model such that the cu... | 8 Gpc | GPQADiamond | 259 | 1 |
A worker receives an annual wage of $\$20{,}000$, which he always deposits into a savings account at the end of the year. By the end of the third year (when he makes the third deposit), he wants to have at least $\$66,200$ in the account to finance the purchase of a house. What is the minimal compound interest rate tha... | 10 | MATH500 | 323 | 1 |
Consider the function \[ f(x) =
\begin{cases}
ax^2 & \text{if } x \geq a,\\
ax +2a& \text{if } x <a,
\end{cases}
\]where $a$ is some number.
What is the largest value of $a$ such that the graph of $y=f(x)$ intersects every horizontal line at least once? | 2 | MATH500 | 387 | 1 |
The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. The line through $G$ that is parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If the area of triangle $ABC$ is 144, then find the area of triangle $ENG$. | 8 | MATH500 | 451 | 1 |
In the land of Ink, the money system is unique. One Trinket is equal to 4 Blinkets, and 3 Blinkets are equal to 7 Drinkets. In Trinkets, what is the value of 56 Drinkets? | 6 | MATH500 | 515 | 1 |
Suppose $a$ and $b$ are different prime numbers greater than 2. How many whole-number divisors are there for the integer $a(2a+b)-2a^{2}+ab$? | 8 | MATH500 | 579 | 1 |
Let \[f(n) =
\begin{cases}
4n+3 &\text{if }n<a, \\
7n-12 &\text{if }n\ge{a}.
\end{cases}
\]Find $a$ if the graph of $y=f(n)$ is continuous. | 5 | MATH500 | 643 | 1 |
A reflection takes $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$ to $\begin{pmatrix} 4 \\ 3 \end{pmatrix}.$ Which vector does the reflection take $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$ to? | \begin{pmatrix} 1/5 \\ -18/5 \end{pmatrix} | MATH500 | 707 | 1 |
At 50 miles per hour, how far would a car travel in $2\frac{3}{4}$ hours? Express your answer as a mixed number. | 137 \frac{1}{2} | MATH500 | 771 | 1 |
You are given a 0-indexed integer array nums representing the score of students in an exam. The teacher would like to form one non-empty group of students with maximal strength, where the strength of a group of students of indices i_0, i_1, i_2, ... , i_k is defined as nums[i_0] * nums[i_1] * nums[i_2] * ... * nums[i_k... | LiveCodeBench | 811 | 1 | |
Given a 0-indexed string s, repeatedly perform the following operation any number of times:
Choose an index i in the string, and let c be the character in position i. Delete the closest occurrence of c to the left of i (if any) and the closest occurrence of c to the right of i (if any).
Your task is to minimize the l... | LiveCodeBench | 832 | 2 | |
You are given a 0-indexed integer array nums. A subarray of nums is called continuous if:
Let i, i + 1, ..., j_ be the indices in the subarray. Then, for each pair of indices i <= i_1, i_2 <= j, 0 <= |nums[i_1] - nums[i_2]| <= 2.
Return the total number of continuous subarrays.
A subarray is a contiguous non-empty se... | LiveCodeBench | 853 | 3 | |
Given a 0-indexed integer array nums of length n and an integer target, return the number of pairs (i, j) where 0 <= i < j < n and nums[i] + nums[j] < target.
Example 1:
Input: nums = [-1,1,2,3,1], target = 2
Output: 3
Explanation: There are 3 pairs of indices that satisfy the conditions in the statement:
- (0, 1) s... | LiveCodeBench | 875 | 1 | |
You are given a 0-indexed 2D integer array nums representing the coordinates of the cars parking on a number line. For any index i, nums[i] = [start_i, end_i] where start_i is the starting point of the i^th car and end_i is the ending point of the i^th car.
Return the number of integer points on the line that are cover... | LiveCodeBench | 896 | 2 | |
You are given a 0-indexed integer array nums of length n.
We want to group the indices so for each index i in the range [0, n - 1], it is assigned to exactly one group.
A group assignment is valid if the following conditions hold:
For every group g, all indices i assigned to group g have the same value in nums.
For an... | LiveCodeBench | 917 | 3 | |
You are given a 0-indexed array nums consisting of positive integers.
A partition of an array into one or more contiguous subarrays is called good if no two subarrays contain the same number.
Return the total number of good partitions of nums.
Since the answer may be large, return it modulo 10^9 + 7.
Example 1:
Inpu... | LiveCodeBench | 939 | 1 | |
You are given three integers start, finish, and limit. You are also given a 0-indexed string s representing a positive integer.
A positive integer x is called powerful if it ends with s (in other words, s is a suffix of x) and each digit in x is at most limit.
Return the total number of powerful integers in the range [... | LiveCodeBench | 960 | 2 | |
An ant is on a boundary. It sometimes goes left and sometimes right.
You are given an array of non-zero integers nums. The ant starts reading nums from the first element of it to its end. At each step, it moves according to the value of the current element:
If nums[i] < 0, it moves left by -nums[i] units.
If nums[i] >... | LiveCodeBench | 981 | 3 | |
On a two-dimensional plane, Takahashi is initially at point (0, 0), and his initial health is H. M items to recover health are placed on the plane; the i-th of them is placed at (x_i,y_i).
Takahashi will make N moves. The i-th move is as follows.
-
Let (x,y) be his current coordinates. He consumes a health of 1 to... | LiveCodeBench | 1,003 | 1 | |
You are given a string S of length N consisting of lowercase English letters and the characters ( and ).
Print the string S after performing the following operation as many times as possible.
- Choose and delete a contiguous substring of S that starts with (, ends with ), and does not contain ( or ) other than the fi... | LiveCodeBench | 1,024 | 2 | |
There is a directed graph with N vertices and N edges.
The i-th edge goes from vertex i to vertex A_i. (The constraints guarantee that i \neq A_i.)
Find a directed cycle without the same vertex appearing multiple times.
It can be shown that a solution exists under the constraints of this problem.
Notes
The sequence ... | LiveCodeBench | 1,045 | 3 | |
We have N books numbered 1 to N.
Book i assumes that you have read C_i books, the j-th of which is book P_{i,j}: you must read all these C_i books before reading book i.
Here, you can read all the books in some order.
You are trying to read the minimum number of books required to read book 1.
Print the numbers of th... | LiveCodeBench | 1,067 | 1 | |
You are given a string S of length N consisting of A, B, and C.
Find the position where ABC first appears as a (contiguous) substring in S. In other words, find the smallest integer n that satisfies all of the following conditions.
- 1 \leq n \leq N - 2.
- The string obtained by extracting the n-th through (n+2)-th ch... | LiveCodeBench | 1,088 | 2 | |
As the factory manager of Keyence, you want to monitor several sections on a conveyor belt. There are a total of N sections you want to monitor, and the length of the i-th section is D_i meters.
There are two types of sensors to choose from, and below is some information about each sensor.
- Type-j sensor (1\leq j \le... | LiveCodeBench | 1,109 | 3 | |
N people labeled 1,2,\dots,N took an exam, and person i scored A_i points.
Only those who scored at least L points pass this exam.
Determine how many people out of the N have passed the exam.
Input
The input is given from Standard Input in the following format:
N L
A_1 A_2 \dots A_N
Output
Print the answer as an... | LiveCodeBench | 1,131 | 1 | |
Takahashi has N pairs of socks, and the i-th pair consists of two socks of color i.
One day, after organizing his chest of drawers, Takahashi realized that he had lost one sock each of colors A_1, A_2, \dots, A_K, so he decided to use the remaining 2N-K socks to make \lfloor\frac{2N-K}{2}\rfloor new pairs of socks, eac... | LiveCodeBench | 1,152 | 2 | |
There is a weighted simple directed graph with N vertices and M edges.
The vertices are numbered 1 to N, and the i-th edge has a weight of W_i and extends from vertex U_i to vertex V_i.
The weights can be negative, but the graph does not contain negative cycles.
Determine whether there is a walk that visits each vert... | LiveCodeBench | 1,173 | 3 | |
There is a simple undirected graph G with N vertices labeled with numbers 1, 2, \ldots, N.
You are given the adjacency matrix (A_{i,j}) of G. That is, G has an edge connecting vertices i and j if and only if A_{i,j} = 1.
For each i = 1, 2, \ldots, N, print the numbers of the vertices directly connected to vertex i in a... | LiveCodeBench | 1,195 | 1 | |
On the xy-plane, there are N points with ID numbers from 1 to N. Point i is located at coordinates (X_i, Y_i), and no two points have the same coordinates.
From each point, find the farthest point and print its ID number.
If multiple points are the farthest, print the smallest of the ID numbers of those points.
Here, ... | LiveCodeBench | 1,216 | 2 | |
The AtCoder railway line has N stations, numbered 1, 2, \ldots, N.
On this line, there are inbound trains that start at station 1 and stop at the stations 2, 3, \ldots, N in order, and outbound trains that start at station N and stop at the stations N - 1, N - 2, \ldots, 1 in order.
Takahashi is about to travel from st... | LiveCodeBench | 1,237 | 3 | |
You are given a 0-indexed array of integers nums of length n, and a positive odd integer k.
The strength of x subarrays is defined as strength = sum[1] * x - sum[2] * (x - 1) + sum[3] * (x - 2) - sum[4] * (x - 3) + ... + sum[x] * 1 where sum[i] is the sum of the elements in the i^th subarray. Formally, strength is sum ... | LiveCodeBench | 1,259 | 1 | |
You are given a string s. s[i] is either a lowercase English letter or '?'.
For a string t having length m containing only lowercase English letters, we define the function cost(i) for an index i as the number of characters equal to t[i] that appeared before it, i.e. in the range [0, i - 1].
The value of t is the sum o... | LiveCodeBench | 1,280 | 2 | |
You are given an array nums consisting of positive integers where all integers have the same number of digits.
The digit difference between two integers is the count of different digits that are in the same position in the two integers.
Return the sum of the digit differences between all pairs of integers in nums.
Ex... | LiveCodeBench | 1,301 | 3 |
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