Found problems: 85335
2021 Ecuador NMO (OMEC), 2
Let $P(x)$ a grade 3 polynomial such that:
$$P(1)=1, P(2)=4, P(3)=9$$
Find the value of $P(10)+P(-6)$
2016 MMPC, 1
If a polygon has both an inscribed circle and a circumscribed circle, then define the [i]halo[/i] of that polygon to be the region inside the circumcircle but outside the incircle. In particular, all regular polygons and all triangles have halos.
(a) What is the area of the halo of a square with side length 2?
(b) What is the area of the halo of a 3-4-5 right triangle?
(c) What is the area of the halo of a regular 2016-gon with side length 2?
2019 USMCA, 28
Alex the Kat plays the following game. First, he writes the number $27000$ on a blackboard. Each minute, he erases the number on the blackboard and replaces it with a number chosen uniformly randomly from its positive divisors, including itself. Find the probability that, after $2019$ minutes, the number on the blackboard is $1$.
2023 South Africa National Olympiad, 6
Let $ABIH$,$BDEC$ and $ACFG$ be arbitrary rectangles constructed (externally) on the sides of triangle $ABC$.Choose point $S$ outside rectangle $ABIH$ (on the opposite side as triangle $ABC$) such that $\angle SHI=\angle FAC$ and $\angle HIS=\angle EBC$.Prove that the lines $FI,EH$ and $CS$ are concurrent(i.e., the three lines intersect in one point).
2020 BAMO, 4
Consider $\triangle ABC$. Choose a point $M$ on side $BC$ and let $O$ be the center of the circle passing through the vertices of $\triangle ABM$. Let $k$ be the circle that passes through $A$ and $M$ and whose center lies on $BC$. Let line $MO$ intersect $K$ again in point $K$. Prove that the line $BK$ is the same for any point $M$ on segment $BC$, so long as all of these constructions are well-defined.
[i]Proposed by Evan Chen[/i]
2018 ITAMO, 5
$5.$Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits.
For instance , $B(\frac{1}{22})={045454,454545,545454}$
Find the minimum number of elements of $B(x)$ as $x$ varies among all irrational numbers with $0<x<1$
2019 IMEO, 2
Consider some graph $G$ with $2019$ nodes. Let's define [i]inverting[/i] a vertex $v$ the following process: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted, and if there wasn't, it is added. We want to minimize the number of edges in the graph by several [i]invertings[/i] (we are allowed to invert the same vertex several times). Find the smallest number $M$ such that we can always make the number of edges in the graph not larger than $M$, for any initial choice of $G$.
[i]Proposed by Arsenii Nikolaev, Anton Trygub (Ukraine)[/i]
2000 JBMO ShortLists, 6
Find all four-digit numbers such that when decomposed into prime factors, each number has the sum of its prime factors equal to the sum of the exponents.
1985 IMO Longlists, 36
Determine whether there exist $100$ distinct lines in the plane having exactly $1985$ distinct points of intersection
2001 China Team Selection Test, 2
Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that:
$\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$
[hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]
1987 Vietnam National Olympiad, 3
Let be given $ n \ge 2$ lines on a plane, not all concurrent and no two parallel. Prove that there is a point which belongs to exactly two of the given lines.
2020 Malaysia IMONST 2, 4
Given are four circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$. Circles $\Gamma_1$ and $\Gamma_2$ are externally tangent at point $A$. Circles $\Gamma_2$ and
$\Gamma_3$ are externally tangent at point $B$. Circles $\Gamma_3$ and $\Gamma_4$ are externally tangent at point $C$. Circles $\Gamma_4$ and
$\Gamma_1$ are externally tangent at point $D$. Prove that $ABCD$ is cyclic.
2015 BMT Spring, 11
Write down $1, 2, 3, ... , 2015$ in a row on a whiteboard. Every minute, select a pair of adjacent numbers at random, erase them, and insert their sum where you selected the numbers. (For instance, selecting $3$ and $4$ from $1, 2, 3, 4, 5$ would result in $1, 2, 7, 5$.) Repeat this process until you have two numbers remaining. What is the probability that the smaller number is less than or equal to $2015$?
1989 AIME Problems, 1
Compute $\sqrt{(31)(30)(29)(28)+1}$.
2012 ELMO Shortlist, 2
Determine whether it's possible to cover a $K_{2012}$ with
a) 1000 $K_{1006}$'s;
b) 1000 $K_{1006,1006}$'s.
[i]David Yang.[/i]
1994 Poland - First Round, 7
(a) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $2f(f(x)) = f(x) \geq 0$ for all $x \in R$.
(b) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $-1 \leq 2f(f(x)) = f(x) \leq 1$ for all $x \in R$.
2023 Regional Olympiad of Mexico Southeast, 1
Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.
2013 BMT Spring, 1
Find the value of $a$ satisfying
\begin{align*}
a+b&=3\\
b+c&=11\\
c+a&=61
\end{align*}
2006 AMC 10, 12
The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$?
$ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$
2024-25 IOQM India, 17
Consider an isosceles triangle $ABC$ with sides $BC = 30, CA = AB = 20$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $M$ be the midpoint of $AD$. Let $PQ$ be a chord of the circumcircle of triangle $ABC$, such that $M$ lies on $PQ$ and $PQ$ is parallel to $BC$. The length of $PQ$ is:
2007 Thailand Mathematical Olympiad, 6
A triangle has perimeter $2s$, inradius $r$, and incenter $I$. If $s_a, s_b$ and $s_c$ are the distances from $I$ to the three vertices, then show that $$\frac34 +\frac{r}{s_a}+\frac{r}{s_b}+\frac{r}{s_c} \le \frac{s^2}{12r^2}$$
2017 Harvard-MIT Mathematics Tournament, 10
Let $ABC$ be a triangle in the plane with $AB = 13$, $BC = 14$, $AC = 15$. Let $M_n$ denote the smallest possible value of $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$ over all points $P$ in the plane. Find $\lim_{n \to \infty} M_n$.
2022 Junior Macedonian Mathematical Olympiad, P3
Let $\triangle ABC$ be an acute triangle with orthocenter $H$. The circle $\Gamma$ with center $H$ and radius $AH$ meets the lines $AB$ and $AC$ at the points $E$ and $F$ respectively. Let $E'$, $F'$ and $H'$ be the reflections of the points $E$, $F$ and $H$ with respect to the line $BC$, respectively. Prove that the points $A$, $E'$, $F'$ and $H'$ lie on a circle.
[i]Proposed by Jasna Ilieva[/i]
2000 239 Open Mathematical Olympiad, 2
100 volleyball teams played a one-round tournament. No two matches held at the same time. It turned out that before each match the teams playing against each other had the same number of wins. Find all possible number of wins for the winner of this tournament.
1965 Poland - Second Round, 6
Prove that there is no polyhedron whose every plane section is a triangle.