Found problems: 85335
2019 Korea Junior Math Olympiad., 1
Each integer coordinates are colored with one color and at least 5 colors are used to color every integer coordinates. Two integer coordinates $(x, y)$ and $(z, w)$ are colored in the same color if $x-z$ and $y-w$ are both multiples of 3. Prove that there exists a line that passes through exactly three points when five points with different colors are chosen randomly.
2010 AMC 10, 13
What is the sum of all the solutions of $ x \equal{} |2x \minus{} |60\minus{}2x\parallel{}$?
$ \textbf{(A)}\ 32\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 124$
2012 Korea Junior Math Olympiad, 3
Find all $l,m,n \in\mathbb{N}$ that satisfies the equation $5^l43^m+1=n^3$
2021 BMT, 21
There exist integers $a$ and $b$ such that $(1 +\sqrt2)^{12}= a + b\sqrt2$. Compute the remainder when $ab$ is divided by $13$.
2013 Serbia Additional Team Selection Test, 3
Let $p > 3$ be a given prime number. For a set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{N}$ , define
$S_a = \{ x \in \{ 0,1, 2,...,p-1 \}$ | $(\exists_s \in S) x \equiv_p a \cdot s \}$ .
$(a)$ How many sets $S \subseteq \{ 1, 2,...,p-1 \} $ are there for which the sequence
$S_1 , S_2 , ..., S_{p-1}$ contains exactly two distinct terms?
$(b)$ Determine all numbers $k \in \mathbb{N}$ for which there is a set $ S \subseteq \{ 1, 2,...,p-1 \} $ such
that the sequence $S_1 , S_2 , ..., S_{p-1} $ contains exactly $k$ distinct terms.
[i]Proposed by Milan Basic and Milos Milosavljevic[/i]
2013 National Olympiad First Round, 30
For how many postive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$?
$
\textbf{(A)}\ 212
\qquad\textbf{(B)}\ 206
\qquad\textbf{(C)}\ 191
\qquad\textbf{(D)}\ 185
\qquad\textbf{(E)}\ 173
$
2015 Princeton University Math Competition, B4
A circle with radius $1$ and center $(0, 1)$ lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of $35$ degrees relative to the positive $x$-axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with $n$ points on the circle. What is $n$?
2023 VIASM Summer Challenge, Problem 4
Let $ABCD$ be a parallelogram and $P$ be an arbitrary point in the plane. Let $O$ be the intersection of two diagonals $AC$ and $BD.$ The circumcircles of triangles $POB$ and $POC$ intersect the circumcircles of triangle $OAD$ at $Q$ and $R,$ respectively $(Q,R \ne O).$ Construct the parallelograms $PQAM$ and $PRDN.$
Prove that: the circumcircle of triangle $MNP$ passes through $O.$
[i]Proposed by Tran Quang Hung ([url=https://artofproblemsolving.com/community/user/68918]buratinogigle[/url])[/i]
2020 Thailand Mathematical Olympiad, 7
Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$.
2019 BMT Spring, 1
A fair coin is repeatedly flipped until $2019$ consecutive coin flips are the same. Compute the probability that the first and last flips of the coin come up differently.