problem stringlengths 76 2.99k | answer stringlengths 0 58 | task stringclasses 6
values | problem_number int64 2 1.3k | set_number int64 1 5 |
|---|---|---|---|---|
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | 113 | AIME24 | 2 | 4 |
Let $A$, $B$, $C$, and $D$ be point on the hyperbola $\frac{x^2}{20}- \frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi. | 480 | AIME24 | 15 | 3 |
Let $N$ be the greatest four-digit positive integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$. Find $Q+R$. | 699 | AIME24 | 28 | 2 |
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 | 1 |
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collec... | 610 | AIME25 | 53 | 5 |
How many complex numbers satisfy the equation $z^5=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$? | 7 | AMC23 | 66 | 4 |
Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$? The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 265 | AMC23 | 79 | 3 |
When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$? | 11 | AMC23 | 92 | 2 |
Compounds that have the same molecular formula but are different in their structural arrangement are known as isomers. Isomers have two types, constitutional isomers and stereoisomers. Constitutional isomers have the same molecular formula but differ in their structures. In stereoisomers, molecules are connected in the... | A = benzoquinone, B = methyl 2-hydroxypropanoate | GPQADiamond | 121 | 1 |
Two planets have been detected orbiting a star using both the transit and radial velocity (RV) methods. The star has a mass of 1.5 times that of the Sun, a radius 1.2 times that of the Sun, and an effective temperature (Teff) of 6300 K. Planet1 has a mass equivalent to 7 Earth masses and a radius 1.7 times that of Eart... | ~ 0.53 | GPQADiamond | 185 | 1 |
You are analyzing a small peptidic compound that has been chemically synthesized. The 1H NMR spectrum of the crude compound appears consistent with the expected molecule, with the exception of two peaks that both correspond to the same alpha-proton. These two peaks have similar chemical shifts and roughly equal integra... | The crude compound exists as a mixture of diastereoisomers | GPQADiamond | 249 | 1 |
Suppose $\sin D = 0.7$ in the diagram below. What is $DE$? [asy]
pair D,E,F;
F = (0,0);
D = (sqrt(51),7);
E = (0,7);
draw(D--E--F--D);
draw(rightanglemark(D,E,F,15));
label("$D$",D,NE);
label("$E$",E,NW);
label("$F$",F,SW);
label("$7$",(E+F)/2,W);
[/asy] | \sqrt{51} | MATH500 | 313 | 1 |
Let $f(x) = x^3 + 3x^2 + 1.$ There exist real numbers $a \neq 0$ and $b,$ such that
\[f(x) - f(a) = (x - a)^2 (x - b).\]Enter the ordered pair $(a,b).$ | (-2,1) | MATH500 | 377 | 1 |
A Senate committee has 5 Democrats, 5 Republicans, and 1 Independent. In how many ways can they sit around a circular table if all the members of each party all sit next to each other? (Two seatings are considered equivalent if one is a rotation of the other.) | 28800 | MATH500 | 441 | 1 |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \end{pmatrix}.\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordered pair $(m,b).$ | \left( \frac{3}{2}, -13 \right) | MATH500 | 505 | 1 |
One ticket to a show costs $\$20$ at full price. Susan buys $4$ tickets using a coupon that gives her a $25\%$ discount. Pam buys $5$ tickets using a coupon that gives her a $30\%$ discount. How many more dollars does Pam pay than Susan? | 10 | MATH500 | 569 | 1 |
If $a+b=8$, $b+c=-3$, and $a+c= -5$, what is the value of the product $abc$? | -120 | MATH500 | 633 | 1 |
Navin from Mauritius, Luka from Croatia, and Ian from Botswana are chatting in the lobby of a youth hostel discussing their part-time jobs. They learn that Navin makes 160 rupee per hour, Luka makes 25 kuna per hour, and Ian makes 34 pula per hour. If one US dollar is equivalent to 32.35 Mauritius rupee, 5.18 Croatian ... | \text{Navin} | MATH500 | 697 | 1 |
Linda, Sherry, June, and Connie walked around their neighborhoods selling girl scout cookies. Linda earned $\$27.47$, Sherry earned $\$35.23$, June earned $\$37.37$, and Connie earned $\$26.52$. After the sale, they pooled their money together and went to the bank to convert the coins into dollars. How much money, in c... | 59 | MATH500 | 761 | 1 |
Yarik is a big fan of many kinds of music. But Yarik loves not only listening to music but also writing it. He likes electronic music most of all, so he has created his own system of music notes, which, in his opinion, is best for it.
Since Yarik also likes informatics, in his system notes are denoted by integers of $... | LiveCodeBench | 807 | 3 | |
You are given a 0-indexed binary string s of length n on which you can apply two types of operations:
Choose an index i and invert all characters from index 0 to index i (both inclusive), with a cost of i + 1
Choose an index i and invert all characters from index i to index n - 1 (both inclusive), with a cost of n - i... | LiveCodeBench | 829 | 1 | |
A truck has two fuel tanks. You are given two integers, mainTank representing the fuel present in the main tank in liters and additionalTank representing the fuel present in the additional tank in liters.
The truck has a mileage of 10 km per liter. Whenever 5 liters of fuel get used up in the main tank, if the addition... | LiveCodeBench | 850 | 2 | |
You are given a 0-indexed integer array nums and a positive integer x.
You are initially at position 0 in the array and you can visit other positions according to the following rules:
If you are currently in position i, then you can move to any position j such that i < j.
For each position i that you visit, you get a ... | LiveCodeBench | 871 | 3 | |
Given a 0-indexed 2D integer matrix grid of size n * m, we define a 0-indexed 2D matrix p of size n * m as the product matrix of grid if the following condition is met:
Each element p[i][j] is calculated as the product of all elements in grid except for the element grid[i][j]. This product is then taken modulo 12345.
... | LiveCodeBench | 893 | 1 | |
You are given a 0-indexed integer array nums.
Return the maximum value over all triplets of indices (i, j, k) such that i < j < k. If all such triplets have a negative value, return 0.
The value of a triplet of indices (i, j, k) is equal to (nums[i] - nums[j]) * nums[k].
Example 1:
Input: nums = [12,6,1,2,7]
Output:... | LiveCodeBench | 914 | 2 | |
You are given three strings s1, s2, and s3. You have to perform the following operation on these three strings as many times as you want.
In one operation you can choose one of these three strings such that its length is at least 2 and delete the rightmost character of it.
Return the minimum number of operations you ne... | LiveCodeBench | 935 | 3 | |
You are given two positive integers x and y.
In one operation, you can do one of the four following operations:
Divide x by 11 if x is a multiple of 11.
Divide x by 5 if x is a multiple of 5.
Decrement x by 1.
Increment x by 1.
Return the minimum number of operations required to make x and y equal.
Example 1:
Inp... | LiveCodeBench | 957 | 1 | |
You are given an array of positive integers nums.
You need to select a subset of nums which satisfies the following condition:
You can place the selected elements in a 0-indexed array such that it follows the pattern: [x, x^2, x^4, ..., x^k/2, x^k, x^k/2, ..., x^4, x^2, x] (Note that k can be be any non-negative power... | LiveCodeBench | 978 | 2 | |
There is an undirected graph with N vertices numbered 1 through N, and initially with 0 edges.
Given Q queries, process them in order. After processing each query,
print the number of vertices that are not connected to any other vertices by an edge.
The i-th query, \mathrm{query}_i, is of one of the following two ki... | LiveCodeBench | 999 | 3 | |
Takahashi has recorded the number of steps he walked for N weeks. He walked A_i steps on the i-th day.
Find the total number of steps Takahashi walked each week.
More precisely, find the sum of the steps for the first week (the 1-st through 7-th day), the sum of the steps for the second week (the 8-th through 14-th da... | LiveCodeBench | 1,021 | 1 | |
We have N dice.
For each i = 1, 2, \ldots, N, when the i-th die is thrown, it shows a random integer between 1 and A_i, inclusive, with equal probability.
Find the probability, modulo 998244353, that the following condition is satisfied when the N dice are thrown simultaneously.
There is a way to choose some (possibl... | LiveCodeBench | 1,042 | 2 | |
You are given a string S consisting of lowercase English letters.
Remove all occurrences of a, e, i, o, u from S and print the resulting string.
S contains at least one character other than a, e, i, o, u.
Input
The input is given from Standard Input in the following format:
S
Output
Print the answer.
Constraints
... | LiveCodeBench | 1,063 | 3 | |
A positive integer x is called a 321-like Number when it satisfies the following condition. This definition is the same as the one in Problem A.
- The digits of x are strictly decreasing from top to bottom.
- In other words, if x has d digits, it satisfies the following for every integer i such that 1 \le i < d:
- (th... | LiveCodeBench | 1,085 | 1 | |
There are zero or more sensors placed on a grid of H rows and W columns. Let (i, j) denote the square in the i-th row from the top and the j-th column from the left.
Whether each square contains a sensor is given by the strings S_1, S_2, \ldots, S_H, each of length W. (i, j) contains a sensor if and only if the j-th ... | LiveCodeBench | 1,106 | 2 | |
You are given a string S of length N consisting of lowercase English letters.
Find the number of non-empty substrings of S that are repetitions of one character. Here, two substrings that are equal as strings are not distinguished even if they are obtained differently.
A non-empty substring of S is a string of length a... | LiveCodeBench | 1,127 | 3 | |
Takahashi will embark on an adventure.
During the adventure, N events will occur.
The i-th event (1\leq i\leq N) is represented by a pair of integers (t _ i,x _ i) (1\leq t _ i\leq 2,1\leq x _ i\leq N) and is as follows:
- If t _ i=1, he finds one potion of type x _ i. He can choose to pick it up or discard it.
- If ... | LiveCodeBench | 1,149 | 1 | |
Your refrigerator has N kinds of ingredients. Let us call them ingredient 1, \dots, ingredient N. You have Q_i grams of ingredient i.
You can make two types of dishes. To make one serving of dish A, you need A_i grams of each ingredient i (1 \leq i \leq N). To make one serving of dish B, you need B_i grams of each ingr... | LiveCodeBench | 1,170 | 2 | |
You are given a string S of length N consisting of lowercase English letters.
You will perform an operation Q times on the string S.
The i-th operation (1\leq i\leq Q) is represented by a pair of characters (c _ i,d _ i), which corresponds to the following operation:
- Replace all occurrences of the character c _ i i... | LiveCodeBench | 1,191 | 3 | |
There is an integer sequence A=(A_1,A_2,\ldots,A_N) of length N, where all elements are initially set to 0. Also, there is a set S, which is initially empty.
Perform the following Q queries in order. Find the value of each element in the sequence A after processing all Q queries. The i-th query is in the following form... | LiveCodeBench | 1,213 | 1 | |
You have an empty sequence and N balls. The size of the i-th ball (1 \leq i \leq N) is 2^{A_i}.
You will perform N operations.
In the i-th operation, you add the i-th ball to the right end of the sequence, and repeat the following steps:
- If the sequence has one or fewer balls, end the operation.
- If the rightmost ... | LiveCodeBench | 1,234 | 2 | |
You are given N intervals of real numbers. The i-th (1 \leq i \leq N) interval is [l_i, r_i]. Find the number of pairs (i, j)\,(1 \leq i < j \leq N) such that the i-th and j-th intervals intersect.
Input
The input is given from Standard Input in the following format:
N
l_1 r_1
l_2 r_2
\vdots
l_N r_N
Output
Prin... | LiveCodeBench | 1,255 | 3 | |
You are given two arrays nums and andValues of length n and m respectively.
The value of an array is equal to the last element of that array.
You have to divide nums into m disjoint contiguous subarrays such that for the i^th subarray [l_i, r_i], the bitwise AND of the subarray elements is equal to andValues[i], in oth... | LiveCodeBench | 1,277 | 1 | |
You are given two strings s and t such that every character occurs at most once in s and t is a permutation of s.
The permutation difference between s and t is defined as the sum of the absolute difference between the index of the occurrence of each character in s and the index of the occurrence of the same character i... | LiveCodeBench | 1,298 | 2 |
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