Found problems: 85335
2006 Balkan MO, 2
Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$ and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point.
[i]Greece[/i]
2022 Iran Team Selection Test, 3
Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$, respectively. Point $X$ is the reflection of $D$ with respect to $B$. Suppose that the line $DE$ is tangent to the $A$-excircle at $Z$. Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$. Prove that the intersection of $BK$ and $AZ$ lies on $\omega$.
Proposed by Mahdi Etesamifard and Alireza Dadgarnia
2011 Sharygin Geometry Olympiad, 5
It is possible to compose a triangle from the altitudes of a given triangle. Can we conclude that it is possible to compose a triangle from its bisectors?
2017 Peru IMO TST, 9
Let $ABCD$ be a cyclie quadrilateral, $\omega$ be it's circumcircle and $M$ be the midpoint of the arc $AB$ of $\omega$ which does not contain the vertices $C$ and $D$. The line that passes through $M$ and the intersection point of segments $AC$ and $BD$, intersects again $\omega$ in $N$. Let $P$ and $Q$ be points in the $CD$ segment such that $\angle AQD = \angle DAP$ and $\angle BPC = \angle CBQ$. Prove that the circumcircle of $NPQ$ and $\omega$ are tangent to each other.
2013 Iran MO (2nd Round), 1
Find all pairs $(a,b)$ of positive integers for which $\gcd(a,b)=1$, and $\frac{a}{b}=\overline{b.a}$. (For example, if $a=92$ and $b=13$, then $b/a=13.92$ )
2001 Mongolian Mathematical Olympiad, Problem 6
In a tennis tournament, any two of the $n$ participants played a match (the winner of a match gets $1$ point, the loser gets $0$). The scores after the tournament were $r_1\le r_2\le\ldots\le r_n$. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set $I=\{i|r_1\ge i\}$. Prove that the number of wrong matches is not less than $\sum_{i\in I}(r_i-i+1)$, and show that this value is realizable
2003 District Olympiad, 1
In the interior of a cube we consider $\displaystyle 2003$ points. Prove that one can divide the cube in more than $\displaystyle 2003^3$ cubes such that any point lies in the interior of one of the small cubes and not on the faces.
2011 Iran MO (3rd Round), 7
Suppose that $f:P(\mathbb N)\longrightarrow \mathbb N$ and $A$ is a subset of $\mathbb N$. We call $f$ $A$-predicting if the set $\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers, it's $A$-predicting.
[i]proposed by Sepehr Ghazi-Nezami[/i]
2003 Cuba MO, 3
A $4 \times 4$ board has all its squares painted white. An allowed operation is to choose a rectangle that contains $3$ squares and paint each of the boxes as follows:
a) If the box is white then it is painted black.
b) If the box is black then it is painted white.
Prove that by applying the allowed operation several times, it is impossible get the entire board painted black.
1978 AMC 12/AHSME, 1
If $1-\frac{4}{x}+\frac{4}{x^2}=0$, then $\frac{2}{x}$ equals
$\textbf{(A) }-1\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }-1\text{ or }2\qquad \textbf{(E) }-1\text{ or }-2$
2018 CCA Math Bonanza, T3
In the game of Avalon, there are $10$ players, $4$ of which are bad. A [i]quest[/i] is a subset of those players. A quest fails if it contains at least one bad player. A randomly chosen quest of $3$ players happens to fail. What is the probability that there is exactly one bad player in the failed quest?
[i]2018 CCA Math Bonanza Team Round #3[/i]
2019 Cono Sur Olympiad, 4
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.
2021 China Team Selection Test, 4
Proof that
$$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$
2008 Harvard-MIT Mathematics Tournament, 8
A piece of paper is folded in half. A second fold is made at an angle $ \phi$ ($ 0^\circ < \phi < 90^\circ$) to the first, and a cut is made as shown below.
[img]12881[/img]
When the piece of paper is unfolded, the resulting hole is a polygon. Let $ O$ be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at $ O$, and also that $ \angle XOY \equal{} 144^\circ$, where $ X$ and $ Y$ are the the vertices of the hole adjacent to $ O$. Find the value(s) of $ \phi$ (in degrees).
2004 Italy TST, 3
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,
\[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]
2017 AMC 8, 25
In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length 2, and $m\angle TUS = 60^\circ$. Arcs $\overarc{TR}$ and $\overarc{SR}$ are each one-sixth of a circle with radius 2. What is the area of the region shown?
[asy]draw((1,1.732)--(2,3.464)--(3,1.732));
draw(arc((0,0),(2,0),(1,1.732)));
draw(arc((4,0),(3,1.732),(2,0)));
label("$U$", (2,3.464), N);
label("$S$", (1,1.732), W);
label("$T$", (3,1.732), E);
label("$R$", (2,0), S);[/asy]
$\textbf{(A) }3\sqrt{3}-\pi\qquad\textbf{(B) }4\sqrt{3}-\frac{4\pi}{3}\qquad\textbf{(C) }2\sqrt{3}\qquad\textbf{(D) }4\sqrt{3}-\frac{2\pi}{3}\qquad\textbf{(E) }4+\frac{4\pi}{3}$
PEN H Problems, 65
Determine all pairs $(x, y)$ of integers such that \[(19a+b)^{18}+(a+b)^{18}+(19b+a)^{18}\] is a nonzero perfect square.
II Soros Olympiad 1995 - 96 (Russia), 9.8
The altitude, angle bisector and median coming from one vertex of the triangle are equal to $\sqrt3$, $2$ and $\sqrt6$, respectively. Find the radius of the circle circumscribed round this triangle.
2016 Iran Team Selection Test, 4
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
2012 AMC 8, 3
On February 13 [i]The Oshkosh Northwester[/i] listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57 \textsc{am}$, and the sunset as $8:15 \textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
$\textbf{(A)}\hspace{.05in}5:10 \textsc{pm} \quad \textbf{(B)}\hspace{.05in}5:21 \textsc{pm} \quad \textbf{(C)}\hspace{.05in}5:41\textsc{pm} \quad \textbf{(D)}\hspace{.05in}5:57 \textsc{pm} \quad \textbf{(E)}\hspace{.05in}6:03 \textsc{pm} $
1983 Brazil National Olympiad, 4
Show that it is possible to color each point of a circle red or blue so that no right-angled triangle inscribed in the circle has its vertices all the same color.
2020 Sharygin Geometry Olympiad, 2
Let $ABCD$ be a cyclic quadrilateral. A circle passing through $A$ and $B$ meets $AC$ and $BD$ at points $E$ and $F$ respectively. The lines $AF$ and $BC$ meet at point $P$, and the lines $BE$ and $AD$ meet at point $Q$. Prove that $PQ$ is parallel to $CD$.
1992 IMO Shortlist, 10
Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$.
[hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]
2021 JHMT HS, 7
In three-dimensional space, let $\mathcal{S}$ be the surface consisting of all points $(x, y, 0)$ satisfying $x^2 + 1 \leq y \leq 2,$ and let $A$ be the point $(0, 0, 900).$ Compute the volume of the solid obtained by taking the union of all line segments with endpoints in $\mathcal{S} \cup \{A\}.$
2017 Dutch IMO TST, 2
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum.
Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)