Found problems: 85335
2014 Ukraine Team Selection Test, 6
Let $n \ge 3$ be an odd integer. Each cell is a $n \times n$ board painted in yellow or blue. Let's call the sequence of cells $S_1, S_2,...,S_m$ [i]path [/i] if they are all the same color and the cells $S_i$ and $S_j$ have one in common an edge if and only if $|i - j| = 1$. Suppose that all yellow cells form a path and all the blue cells form a path. Prove that one of the two paths begins or ends at the center of the board.
2013 Saudi Arabia BMO TST, 8
Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.
2000 Harvard-MIT Mathematics Tournament, 35
If $1+2x+3x^2 +...=9$, find $x$.
2011 Switzerland - Final Round, 9
For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]
2010 LMT, 12
$a,b,c,d,e$ are equal to $1,2,3,4,5$ in some order, such that no two of $a,b,c,d,e$ are equal to the same integer. Given that $b \leq d, c \geq a,a \leq e,b \geq e,$ and that $d\neq5,$ determine the value of $a^b+c^d+e.$
2008 Romania Team Selection Test, 3
Let $ ABCDEF$ be a convex hexagon with all the sides of length 1. Prove that one of the radii of the circumcircles of triangles $ ACE$ or $ BDF$ is at least 1.
2018 HMNT, 3
$HOW,BOW,$ and $DAH$ are equilateral triangles in a plane such that $WO=7$ and $AH=2$. Given that $D,A,B$ are collinear in that order, find the length of $BA$.
1992 IMO Longlists, 56
A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If $x, u,$ and $v$ are three distinct vertices such that $x \to u$ and $x \to v$, then $u \to w$ and $v \to w$ for some vertex $w$. Suppose that $x \to u \to y \to\cdots \to z$ is a path of length $n$, that cannot be extended to the right (no arrow goes away from $z$). Prove that every path beginning at $x$ arrives after $n$ steps at $z.$
1953 Moscow Mathematical Olympiad, 251
On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?
2023 Austrian MO National Competition, 3
Given a positive integer $n$, find the proportion of the subsets of $\{1,2, \ldots, 2n\}$ such that their smallest element is odd.
2011 Argentina Team Selection Test, 5
At least $3$ players take part in a tennis tournament. Each participant plays exactly one match against each other participant. After the tournament has ended, we find out that each player has won at least one match. (There are no ties in tennis).
Show that in the tournament, there was at least one trio of players $A,B,C$ such that $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.
2013 Taiwan TST Round 1, 4
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?
2021-IMOC qualification, A3
Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$
1994 Mexico National Olympiad, 5
$ABCD$ is a convex quadrilateral. Take the $12$ points which are the feet of the altitudes in the triangles $ABC, BCD, CDA, DAB$. Show that at least one of these points must lie on the sides of $ABCD$.
2022 Malaysia IMONST 2, 1
Given a circle and a quadrilateral $ABCD$ whose vertices all lie on the circle. Let $R$ be the midpoint of arc $AB$. The line $RC$ meets line $AB$ at point $S$, and the line $RD$ meets line $AB$ at point $T$.
Prove that $CDTS$ is a cyclic quadrilateral.
2020 IMO Shortlist, A7
Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has
\[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]
2016 Regional Olympiad of Mexico West, 2
Let $A$ be an infinite set of real numbers containing at least one irrational number. Prove that for every natural number $n > 1$ there exists a subset $S$ of $A$ with n elements such that the sum of the elements of $S$ is an irrational number.
2011 AIME Problems, 15
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.
2014 Junior Balkan Team Selection Tests - Moldova, 7
Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$. Determine the measure of the angle $CBF$.
1989 Cono Sur Olympiad, 2
Find the sum\[1+11+111+\cdots+\underbrace{111\ldots111}_{n\text{ digits}}.\]
2010 Laurențiu Panaitopol, Tulcea, 1
Solve in the real numbers the equation $ \arcsin x=\lfloor 2x \rfloor . $
[i]Petre Guțescu[/i]
2020 Federal Competition For Advanced Students, P2, 2
In the plane there are $2020$ points, some of which are black and the rest are green. For every black point, the following applies: [i]There are exactly two green points that represent the distance $2020$ from that black point. [/i]
Find the smallest possible number of green dots.
(Walther Janous)
2005 District Olympiad, 1
Prove that for all $a\in\{0,1,2,\ldots,9\}$ the following sum is divisible by 10:
\[ S_a = \overline{a}^{2005} + \overline{1a}^{2005} + \overline{2a}^{2005} + \cdots + \overline{9a}^{2005}. \]
2020 Jozsef Wildt International Math Competition, W35
In all triangles $ABC$ does it hold:
$$(b^n+c^p)\tan^{n+p}\frac A2+(c^n+a^p)\tan^{n+p}\frac B2+(a^n+b^p)\tan^{n+p}\frac C2\ge6\sqrt{\left(\frac{4r^2}{R\sqrt3}\right)^{n+p}}$$
where $n,p\in(0,\infty)$.
[i]Proposed by Nicolae Papacu[/i]
2020 Balkan MO, 4
Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$.
[i] Proposed by Ilija Jovčevski, North Macedonia[/i]