problem stringlengths 44 3.58k | answer stringlengths 0 131 | 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 | 2 |
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 | 1 |
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 | 5 |
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 | 4 |
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 | 3 |
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 | 2 |
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 | 1 |
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 | 5 |
You have an interesting drought-resistant cultivar of barley, which, unfortunately, contains an anti-nutritional compound. This compound makes it impossible to use this cultivar for food purposes. After an extensive investigation, you discover the gene responsible for the synthesis of this compound. This gene consists ... | Mutant 2 | GPQADiamond | 129 | 1 |
"Scientist aims to run quantitative polymerase chain reaction (qPCR) on some samples. He first wants to make a calibration curve so he took positive control with known copy numbers and run qPCR on serially diluted positive control three technical replicates. The efficiency of qPCR was 100%. The R2 was 1, and the slope ... | Ct values are not in agreement with the amount of target nucleic acid in samples | GPQADiamond | 193 | 1 |
There has been an outbreak of an viral infectious disease in your city. You have been asked to design a molecular diagnostic kit for quick detection of this retrovirus. How would you go about designing this kit? | First you identify the virus by performing cDNA sequencing. You then develop a real time PCR kit for quick and accurate diagnosis. | GPQADiamond | 257 | 1 |
Denali and Nate work for a dog walking business and are paid for each dog they walk. Denali is responsible for $16$ dogs and Nate is responsible for $12$ dogs. Under the company's new policy, they will be assigned or unassigned new dogs in groups of $x$ dogs. The ratio of Denali's pay to Nate's pay would be the same if... | 5 | MATH500 | 321 | 1 |
You have 5 shirts, 6 pairs of pants, and 8 hats. How many outfits can you make consisting of one shirt, one pair of pants, and one hat? | 240 | MATH500 | 385 | 1 |
For every positive integer $n$, let $\text{mod}_5 (n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively as follows:
\[f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1... | 1 | MATH500 | 449 | 1 |
There are 360 people in my school. 15 take calculus, physics, and chemistry, and 15 don't take any of them. 180 take calculus. Twice as many students take chemistry as take physics. 75 take both calculus and chemistry, and 75 take both physics and chemistry. Only 30 take both physics and calculus. How many studen... | 110 | MATH500 | 513 | 1 |
What's the largest eight-digit base 2 integer? Express your answer in base 10. | 255 | MATH500 | 577 | 1 |
What integer $x$ satisfies $\frac{1}{4}<\frac{x}{7}<\frac{1}{3}$? | 2 | MATH500 | 641 | 1 |
What is the remainder of 2004 divided by 12? | 0 | MATH500 | 705 | 1 |
Let $a,$ $b,$ $c,$ and $d$ be positive real numbers such that $a + b + c + d = 10.$ Find the maximum value of $ab^2 c^3 d^4.$ | 27648 | MATH500 | 769 | 1 |
You are given a 0-indexed integer array nums of length n and an integer k. In an operation, you can choose an element and multiply it by 2.
Return the maximum possible value of nums[0] | nums[1] | ... | nums[n - 1] that can be obtained after applying the operation on nums at most k times.
Note that a | b denotes the bi... | LiveCodeBench | 810 | 2 | |
You are given an integer n that consists of exactly 3 digits.
We call the number n fascinating if, after the following modification, the resulting number contains all the digits from 1 to 9 exactly once and does not contain any 0's:
Concatenate n with the numbers 2 * n and 3 * n.
Return true if n is fascinating, or f... | LiveCodeBench | 831 | 3 | |
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 | 1 | |
You are given an array nums of length n and an integer m. You need to determine if it is possible to split the array into n non-empty arrays by performing a series of steps.
In each step, you can select an existing array (which may be the result of previous steps) with a length of at least two and split it into two sub... | LiveCodeBench | 874 | 2 | |
You are given two 0-indexed binary strings s1 and s2, both of length n, and a positive integer x.
You can perform any of the following operations on the string s1 any number of times:
Choose two indices i and j, and flip both s1[i] and s1[j]. The cost of this operation is x.
Choose an index i such that i < n - 1 and f... | LiveCodeBench | 895 | 3 | |
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 | 1 | |
You are given a 0-indexed integer array nums.
You can perform any number of operations, where each operation involves selecting a subarray of the array and replacing it with the sum of its elements. For example, if the given array is [1,3,5,6] and you select subarray [3,5] the array will convert to [1,8,6].
Return the ... | LiveCodeBench | 938 | 2 | |
You are given an array nums consisting of positive integers.
Return the total frequencies of elements in nums such that those elements all have the maximum frequency.
The frequency of an element is the number of occurrences of that element in the array.
Example 1:
Input: nums = [1,2,2,3,1,4]
Output: 4
Explanation: T... | LiveCodeBench | 959 | 3 | |
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 | 1 | |
N people numbered 1,2,\ldots,N were in M photos. In each of the photos, they stood in a single line. In the i-th photo, the j-th person from the left is person a_{i,j}.
Two people who did not stand next to each other in any of the photos may be in a bad mood.
How many pairs of people may be in a bad mood? Here, we... | LiveCodeBench | 1,002 | 2 | |
Takahashi has two sheets A and B, each composed of black squares and transparent squares, and an infinitely large sheet C composed of transparent squares.
There is also an ideal sheet X for Takahashi composed of black squares and transparent squares.
The sizes of sheets A, B, and X are H_A rows \times W_A columns, H_B ... | LiveCodeBench | 1,023 | 3 | |
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 | 1 | |
There are H \times W cookies in H rows and W columns.
The color of the cookie at the i-row from the top and j-th column from the left is represented by a lowercase English letter c_{i,j}.
We will perform the following procedure.
1. For each row, perform the following operation: if there are two or more cookies remain... | LiveCodeBench | 1,066 | 2 | |
There is a tree with N vertices numbered 1 to N.
For each i\ (2 \leq i \leq N), there is an edge connecting vertex i and vertex \lfloor \frac{i}{2} \rfloor.
There are no other edges.
In this tree, find the number of vertices whose distance from vertex X is K.
Here, the distance between two vertices u and v is define... | LiveCodeBench | 1,087 | 3 | |
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 | 1 | |
There are N boxes numbered 1, 2, \ldots, N. Initially, box i contains one ball of color C_i.
You are given Q queries, which you should process in order.
Each query is given by a pair of integers (a,b) and asks you to do the following:
- Move all the balls from box a to box b, and then print the number of different col... | LiveCodeBench | 1,130 | 2 | |
There is a road that stretches infinitely to the east and west, and the coordinate of a point located x meters to the east from a certain reference point on this road is defined as x.
In particular, the coordinate of a point located x meters to the west from the reference point is -x.
Snuke will set up Christmas trees ... | LiveCodeBench | 1,151 | 3 | |
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 | 1 | |
You are given two integers A and B, each between 0 and 9, inclusive.
Print any integer between 0 and 9, inclusive, that is not equal to A + B.
Input
The input is given from Standard Input in the following format:
A B
Output
Print any integer between 0 and 9, inclusive, that is not equal to A + B.
Constraints
- 0... | LiveCodeBench | 1,194 | 2 | |
There are N types of beans, one bean of each type. The i-th type of bean has a deliciousness of A_i and a color of C_i. The beans are mixed and can only be distinguished by color.
You will choose one color of beans and eat one bean of that color. By selecting the optimal color, maximize the minimum possible deliciousne... | LiveCodeBench | 1,215 | 3 | |
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 | 1 | |
You are given an array arr of size n consisting of non-empty strings.
Find a string array answer of size n such that:
answer[i] is the shortest substring of arr[i] that does not occur as a substring in any other string in arr. If multiple such substrings exist, answer[i] should be the lexicographically smallest. And i... | LiveCodeBench | 1,258 | 2 | |
You are given a 0-indexed array nums of size n consisting of positive integers.
You are also given a 2D array queries of size m where queries[i] = [index_i, k_i].
Initially all elements of the array are unmarked.
You need to apply m queries on the array in order, where on the i^th query you do the following:
Mark the ... | LiveCodeBench | 1,279 | 3 | |
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 | 1 |
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