Found problems: 85335
2011 Croatia Team Selection Test, 3
Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.
2020 AIME Problems, 9
While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.
1975 Bulgaria National Olympiad, Problem 6
Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$.
[i](H. Lesov)[/i]
2018 Oral Moscow Geometry Olympiad, 6
Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.
2009 ITAMO, 1
A flea is initially at the point $(0, 0)$ in the Cartesian plane. Then it makes $n$ jumps. The direction of the jump is taken in a choice of the four cardinal directions. The first step is of length $1$, the second of length $2$, the third of length $4$, and so on. The $n^{th}$-jump is of length $2^{n-1}$. Prove that, if you know the final position flea, then it is possible to uniquely determine its position after each of the $n$ jumps.
2021 Indonesia TST, G
Given points $A$, $B$, $C$, and $D$ on circle $\omega$ such that lines $AB$ and $CD$ intersect on point $T$ where $A$ is between $B$ and $T$, moreover $D$ is between $C$ and $T$. It is known that the line passing through $D$ which is parallel to $AB$ intersects $\omega$ again on point $E$ and line $ET$ intersects $\omega$ again on point $F$. Let $CF$ and $AB$ intersect on point $G$, $X$ be the midpoint of segment $AB$, and $Y$ be the reflection of point $T$ to $G$.
Prove that $X$, $Y$, $C$, and $D$ are concyclic.
2011 Singapore MO Open, 3
Let $x,y,z>0$ such that $\frac1x+\frac1y+\frac1z<\frac{1}{xyz}$. Show that
\[\frac{2x}{\sqrt{1+x^2}}+\frac{2y}{\sqrt{1+y^2}}+\frac{2z}{\sqrt{1+z^2}}<3.\]
2009 Iran Team Selection Test, 11
Let $n$ be a positive integer. Prove that \[ 3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}. \]
1980 USAMO, 3
Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.
2003 Junior Tuymaada Olympiad, 8
A few people came to the party. Prove that they can be placed in two rooms so that each of them has in their own room an even number of acquaintances. (One of the rooms can be left empty.)
2008 Grigore Moisil Intercounty, 1
On a circle there are given $ n\plus{}3$ distinct points,from which $ n$ are colored red, two yellow, and one blue. Determine the number of polygons which have
a) the vertices of the same color
b) the vertices of two colors
c) the vertices of three colors.
2007 Iran Team Selection Test, 3
Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence.
a) Prove that $B_{n}$ does not depend on location of $P$.
b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.
1998 Mexico National Olympiad, 6
A plane in space is equidistant from a set of points if its distances from the points in the set are equal. What is the largest possible number of equidistant planes from five points, no four of which are coplanar?
2000 Romania Team Selection Test, 3
Determine all pairs $(m,n)$ of positive integers such that a $m\times n$ rectangle can be tiled with L-trominoes.
2010 Today's Calculation Of Integral, 551
In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$.
2010 Peru Iberoamerican Team Selection Test, P6
On an $n$ × $n$ board, the set of all squares that are located on or below the main diagonal of the board is called the$n-ladder$. For example, the following figure shows a $3-ladder$:
[asy]
draw((0,0)--(0,3));
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((1,0)--(1,3));
draw((0,2)--(2,2));
draw((2,0)--(2,2));
draw((0,3)--(1,3));
draw((3,0)--(3,1));
[/asy]
In how many ways can a $99-ladder$ be divided into some rectangles, which have their sides on grid lines, in such a way that all the rectangles have distinct areas?
2019 USEMO, 3
Consider an infinite grid $\mathcal G$ of unit square cells. A [i]chessboard polygon[/i] is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$.
Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible?
[i]Nikolai Beluhov[/i]
2009 Turkey MO (2nd round), 1
Let $H$ be the orthocenter of an acute triangle $ABC,$ and let $A_1, \: B_1, \: C_1$ be the feet of the altitudes belonging to the vertices $A, \: B, \: C,$ respectively. Let $K$ be a point on the smaller $AB_1$ arc of the circle with diameter $AB$ satisfying the condition $\angle HKB = \angle C_1KB.$ Let $M$ be the point of intersection of the line segment $AA_1$ and the circle with center $C$ and radius $CL$ where $KB \cap CC_1=\{L\}.$ Let $P$ and $Q$ be the points of intersection of the line $CC_1$ and the circle with center $B$ and radius $BM.$ Show that $A, \: K, \: P, \: Q$ are concyclic.
2014 South East Mathematical Olympiad, 3
Let $p$ be a primes ,$x,y,z $ be positive integers such that $x<y<z<p$ and $\{\frac{x^3}{p}\}=\{\frac{y^3}{p}\}=\{\frac{z^3}{p}\}$.
Prove that $(x+y+z)|(x^5+y^5+z^5).$
2024 Harvard-MIT Mathematics Tournament, 2
Let $ABC$ be a triangle with $\angle BAC = 90^o$. Let $D$, $E$, and $F$ be the feet of altitude, angle bisector, and median from $A$ to $BC$, respectively. If $DE = 3$ and $EF = 5$, compute the length of $BC$.
2024 HMNT, 32
Let $ABC$ be an acute triangle and $D$ be the foot of altitude from $A$ to $BC.$ Let $X$ and $Y$ be points on the segment $BC$ such that $\angle{BAX} = \angle{YAC}, BX = 2, XY = 6,$ and $YC = 3.$ Given that $AD = 12,$ compute $BD.$
2010 Contests, 3
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
2015 South East Mathematical Olympiad, 6
In $\triangle ABC$, we have three edges with lengths $BC=a, \, CA=b \, AB=c$, and $c<b<a<2c$. $P$ and $Q$ are two points of the edges of $\triangle ABC$, and the straight line $PQ$ divides $\triangle ABC$ into two parts with the same area. Find the minimum value of the length of the line segment $PQ$.
2021 LMT Fall, 13
Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$.
Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively prime to $i$.
2017 Portugal MO, 6
In a building whose floors are numbered $1$ to $8$, the builder wants to place elevators so that, for every choice of two floors, there are always at least three elevators that stop on those floors. Furthermore, each elevator can only stop at a maximum of $5$ floors. What is the minimum number of elevators that need to be placed?