This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 30

2024 Princeton University Math Competition, 15
Tags: Team Round
There are $10$ teams, named $T_1$ through $T_{10},$ participating in a draft in which there are $20$ players available, named $P_1$ through $P_{20}.$ Suppose each team independent of the others has uniform random preference on the $20$ players. Team $T_1$ will draft their favorite player, and then each subsequent team $T_2, \ldots , T_{10}$ draft their favorite player among the ones not already drafted. Each team drafts exactly one player. Given that $P_1$ is among the $10$ favorite players for each team, the probability that $P_1$ is drafted can be written as $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $m + n.$

2018 PUMaC Team Round, 3
Tags: PuMAC , Team Round
The value of $$\frac{\log_35\log_25}{\log_35+\log_25}$$ can be expressed as $a\log_bc$, where $a$, $b$, and $c$ are positive integers, and $a+b$ is as small as possible. Find $a+2b+3c$.

2018 PUMaC Team Round, 14
Tags: PuMAC , Team Round , geometry , 3D geometry
Find the sum of the positive integer solutions to the equation $\left\lfloor\sqrt[3]{x}\right\rfloor+\left\lfloor\sqrt[4]{x}\right\rfloor=4.$

2018 PUMaC Team Round, 10
Tags: PuMAC , Team Round
For how many ordered quadruplets $(a,b,c,d)$ of positive integers such that $2\leq a\leq b \leq c$ and $1 \leq d \leq 418$ do we have that $bcd+abd+acd=abc+abcd?$

2018 PUMaC Team Round, 8
Tags: PuMAC , Team Round
Jackson has a $5\times 5$ grid of squares. He places coins in the grid squares $-$ at most one per square $-$ so that no row, column, or diagonal has five coins. What is the maximum number of coins that he can place?