Found problems: 85335
2000 Croatia National Olympiad, Problem 2
Let $ABC$ be a triangle with $AB = AC$. With center in a point of the side $BC$, the circle $S$ is constructed that is tangent to the sides $AB$ and $AC$. Let $P$ and $Q$ be any points on the sides $AB$ and $AC$ respectively, such that $PQ$ is tangent to $S$. Show that $PB \cdot CQ = \left(\frac{BC}{2}\right)^2$
2023 Abelkonkurransen Finale, 1b
In the triangle $ABC$, points $D$ and $E$ lie on the side $BC$, with $CE = BD$. Also, $M$ is the midpoint of $AD$. Show that the centroid of $ABC$ lies on $ME$.
2009 IMO Shortlist, 6
Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent.
[i]Proposed by Eugene Bilopitov, Ukraine[/i]
2017 Saint Petersburg Mathematical Olympiad, 7
Divide the upper right quadrant of the plane into square cells with side length $1$. In this quadrant, $n^2$ cells are colored, show that there’re at least $n^2+n$ cells (possibly including the colored ones) that at least one of its neighbors are colored.
2005 China Team Selection Test, 3
Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
2019 Romania EGMO TST, P4
Six boys and six girls are participating at a tango course. They meet every evening for three weeks (a total of 21 times). Each evening, at least one boy-girl pair is selected to dance in front of the others. At the end of the three weeks, every boy-girl pair has been selected at least once. Prove that there exists a person who has been selected on at least 5 distinct evenings.
[i]Note:[/i] a person can be selected twice on the same evening.
1989 Federal Competition For Advanced Students, P2, 3
Show that it is possible to situate eight parallel planes at equal distances such that each plane contains precisely one vertex of a given cube. How many such configurations of planes are there?
2024 Belarus - Iran Friendly Competition, 1.2
Given $n \geq 2$ positive real numbers $x_1 \leq x_2 \leq \ldots \leq x_n$ satisfying the equalities
$$x_1+x_2+\ldots+x_n=4n$$
$$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}=n$$
Prove that $\frac{x_n}{x_1} \geq 7+4\sqrt{3}$
2011 LMT, 7
A triangle $ABC$ has side lengths $AB=8$ and $BC=10.$ Given that the altitude to side $BC$ has length $4,$ what is the length of the altitude to side $AB?$
2022 Polish Junior Math Olympiad Second Round, 5.
Let $n\geq 3$ be an odd integer. On a line, $n$ points are marked in such a way that the distance between any two of them is an integer. It turns out that each marked point has an even sum of distances to the remaining $n-1$ marked points. Prove that the distance between any two marked points is even.
2024 Thailand TST, 1
Determine all polynomials $P$ with integer coefficients for which there exists an integer $a_n$ such that $P(a_n)=n^n$ for all positive integers $n$.
2019 CCA Math Bonanza, TB4
The number $28!$ ($28$ in decimal) has base $30$ representation \[28!=Q6T32S??OCLQJ6000000_{30}\] where the seventh and eighth digits are missing. What are the missing digits? In base $30$, we have that the digits $A=10$, $B=11$, $C=12$, $D=13$, $E=14$, $F=15$, $G=16$, $H=17$, $I=18$, $J=19$, $K=20$, $L=21$, $M=22$, $N=23$, $O=24$, $P=25$, $Q=26$, $R=27$, $S=28$, $T=29$.
[i]2019 CCA Math Bonanza Tiebreaker Round #4[/i]
2022 Girls in Math at Yale, 6
Carissa is crossing a very, very, very wide street, and did not properly check both ways before doing so. (Don't be like Carissa!) She initially begins walking at $2$ feet per second. Suddenly, she hears a car approaching, and begins running, eventually making it safely to the other side, half a minute after she began crossing. Given that Carissa always runs $n$ times as fast as she walks and that she spent $n$ times as much time running as she did walking, and given that the street is $260$ feet wide, find Carissa's running speed, in feet per second.
[i]Proposed by Andrew Wu[/i]
2014 Contests, 3
Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.
Durer Math Competition CD Finals - geometry, 2018.C+2
Given an $ABC$ triangle. Let $D$ be an extension of section $AB$ beyond $A$ such that that $AD = BC$ and $E$ is the extension of the section $BC$ beyond $B$ such that $BE = AC$. Prove that the circumcircle of triangle $DEB$ passes through the center of the inscribed circle of triangle $ABC$.
1963 Miklós Schweitzer, 8
Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be
absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]
2008 Indonesia TST, 2
Let $S = \{1, 2, 3, ..., 100\}$ and $P$ is the collection of all subset $T$ of $S$ that have $49$ elements, or in other words: $$P = \{T \subset S : |T| = 49\}.$$ Every element of $P$ is labelled by the element of $S$ randomly (the labels may be the same). Show that there exist subset $M$ of $S$ that has $50$ members such that for every $x \in M$, the label of $M -\{x\}$ is not equal to $x$
2023 Polish MO Finals, 2
Given an acute triangle $ABC$ with their incenter $I$. Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$. Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$. It is given that $$\angle AIX = \angle XYA = 120^\circ.$$
Prove that $YI$ is the angle bisector of $XYA$.
1981 Austrian-Polish Competition, 1
Find the smallest $n$ for which we can find $15$ distinct elements $a_{1},a_{2},...,a_{15}$ of $\{16,17,...,n\}$ such that $a_{k}$ is a multiple of $k$.
1982 IMO Longlists, 40
We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a $3k \times n$ rectangle, then it is impossible to reach a position with only one pawn on the board.
2018 PUMaC Team Round, 5
There exist real numbers $a$, $b$, $c$, $d$, and $e$ such that for all positive integers $n$, we have
$$\sqrt{n}=\sum_{i=0}^{n-1}\sqrt[5]{\sqrt{ai^5+bi^4+ci^3+di^2+ei+1}-\sqrt{ai^5+bi^4+ci^3+di^2+ei}}.$$
Find $a+b+c+d$.
1991 China National Olympiad, 5
Find all natural numbers $n$, such that $\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991$. ($[n/k^2]$ denotes the integer part of $n/k^2$.)
2011 Junior Balkan MO, 1
Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that:
$\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$
2015 Portugal MO, 6
For what values of $n$ is it possible to mark $n$ points on the plane so that each point has at least three other points at distance $1$?
2007 Mongolian Mathematical Olympiad, Problem 4
Let $ a,b,c>0$. Prove that
$ \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}$