Found problems: 85335
2007 IMO, 2
Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$.
[i]Author: Charles Leytem, Luxembourg[/i]
2015 Korea - Final Round, 3
There are at least $3$ subway stations in a city.
In this city, there exists a route that passes through more than $L$ subway stations, without revisiting.
Subways run both ways, which means that if you can go from subway station A to B, you can also go from B to A.
Prove that at least one of the two holds.
$\text{(i)}$. There exists three subway stations $A$, $B$, $C$ such that there does not exist a route from $A$ to $B$ which doesn't pass through $C$.
$\text{(ii)}$. There is a cycle passing through at least $\lfloor \sqrt{2L} \rfloor$ stations, without revisiting a same station more than once.
2023 Pan-American Girls’ Mathematical Olympiad, 4
In an acute-angled triangle $ABC$, let $D$ be a point on the segment $BC$. Let $R$ and $S$ be the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. The line $DR$ intersects the circumcircle of $BDS$ at $X$, with $X \neq D$. Similarly, the line $DS$ intersects the circumcircle of $CDR$ at $Y$, with $Y \neq D$. Prove that if $XY$ is parallel to $RS$, then $D$ is the midpoint of $BC$.
2008 Bulgarian Autumn Math Competition, Problem 11.2
On the sides $AB$ and $AC$ of the right $\triangle ABC$ ($\angle A=90^{\circ}$) are chosen points $C_{1}$ and $B_{1}$ respectively. Prove that if $M=CC_{1}\cap BB_{1}$ and $AC_{1}=AB_{1}=AM$, then $[AB_{1}MC_{1}]+[AB_{1}C_{1}]=[BMC]$.
KoMaL A Problems 2023/2024, A. 868
A set of points in the plane is called disharmonic, if the ratio of any two distances between the points is between $100/101$ and $101/100$, or at least $100$ or at most $1/100$.
Is it true that for any distinct points $A_1,A_2,\ldots,A_n$ in the plane it is always possible to find distinct points $A_1',A_2',\ldots, A_n'$ that form a disharmonic set of points, and moreover $A_i, A_j$ and $A_k$ are collinear in this order if and only if $A_i', A_j'$ and $A_k'$ are collinear in this order (for all distinct $1 \le i,j,k\le n$?
[i]Submitted by Dömötör Pálvölgyi and Balázs Keszegh, Budapest[/i]
1992 Bulgaria National Olympiad, Problem 1
Through a random point $C_1$ from the edge $DC$ of the regular tetrahedron $ABCD$ is drawn a plane, parallel to the plane $ABC$. The plane constructed intersects the edges $DA$ and $DB$ at the points $A_1,B_1$ respectively. Let the point $H$ is the midpoint of the altitude through the vertex $D$ of the tetrahedron $DA_1B_1C_1$ and $M$ is the center of gravity (barycenter) of the triangle $ABC_1$. Prove that the measure of the angle $HMC$ doesn’t depend on the position of the point $C_1$. [i](Ivan Tonov)[/i]
2011 IMO Shortlist, 3
Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$
[i]Proposed by Mihai Baluna, Romania[/i]
2007 Purple Comet Problems, 20
Three congruent ellipses are mutually tangent. Their major axes are parallel. Two of the ellipses are tangent at the end points of their minor axes as shown. The distance between the centers of these two ellipses is $4$. The distances from those two centers to the center of the third ellipse are both $14$. There are positive integers m and n so that the area between these three ellipses is $\sqrt{n}-m \pi$. Find $m+n$.
[asy]
size(250);
filldraw(ellipse((2.2,0),2,1),grey);
filldraw(ellipse((0,-2),4,2),white);
filldraw(ellipse((0,+2),4,2),white);
filldraw(ellipse((6.94,0),4,2),white);[/asy]
2016 Saudi Arabia Pre-TST, 1.3
Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.
1972 IMO Longlists, 4
You have a triangle, $ABC$. Draw in the internal angle trisectors. Let the two trisectors closest to $AB$ intersect at $D$, the two trisectors closest to $BC$ intersect at $E$, and the two closest to $AC$ at $F$. Prove that $DEF$ is equilateral.
LMT Guts Rounds, 2020 F32
In a lottery there are $14$ balls, numbered from $1$ to $14$. Four of these balls are drawn at random. D'Angelo wins the lottery if he can split the four balls into two disjoint pairs, where the two balls in each pair have difference at least $5$. The probability that D'Angelo wins the lottery can be expressed as $\frac{m}{n}$, with $m,n$ relatively prime. Find $m+n$.
[i]Proposed by Richard Chen[/i]
1995 Israel Mathematical Olympiad, 2
Let $PQ$ be the diameter of semicircle $H$. Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$. Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB\perp PQ$ and is tangent to $O$. Prove that $AC$ bisects $\angle PAB$
2016 Costa Rica - Final Round, N3
Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$
2023 Balkan MO Shortlist, C1
Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?
2014 Kyiv Mathematical Festival, 2
Can an $8\times8$ board be covered with 13 equal 5-celled figures?
It's alowed to rotate the figures or turn them over.
[size=85](Kyiv mathematical festival 2014)[/size]
2004 District Olympiad, 1
If reals $a,b,c$ satisfy $a^2+b^2+c^2=3$ then prove that $|a|+|b|+|c|-abc\leq4$.
1957 Putnam, A6
Let $a>0$, $S_1 =\ln a$ and $S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )$ for $n >1.$ Show that
$$ \lim_{n \to \infty} S_n = a-1.$$
2024 Belarus Team Selection Test, 1.2
An acute-angled triangle $ABC$ with an altitude $AD$ and orthocenter $H$ are given. $AD$ intersects the circumcircle of $ABC$ $\omega$ at $P$. $K$ is a point on segment $BC$ such that $KC=BD$. The circumcircle of $KPH$ intersects $\omega$ at $Q$ and $BC$ at $N$. A line perpendicular to $PQ$ and passing through $N$ intersects $AD$ at $T$. Prove that the center of $\omega$ lies on line $TK$.
[i]U. Maksimenkau[/i]
1997 Romania National Olympiad, 3
$ABCDA'B'CD'$ is a rectangular parallelepiped with $AA'= 2AB = 8a$ , $E$ is the midpoint of $(AB)$ and $M$ is the point of $(DD')$ for which $DM = a \left( 1 + \frac{AD}{AC}\right)$.
a) Find the position of the point. $F$ on the segment $(AA')$ for which the sum $CF + FM$ has the minimum possible value.
b) Taking $F$ as above, compute the measure of the angle of the planes $(D, E, F)$ and $(D, B', C')$.
c) Knowing that the straight lines $AC'$ and $FD$ are perpendicular, compute the volume of the parallelepiped $ABCDA'B'C'D'$.
2002 Irish Math Olympiad, 3
Find all triples of positive integers $ (p,q,n)$, with $ p$ and $ q$ primes, satisfying:
$ p(p\plus{}3)\plus{}q(q\plus{}3)\equal{}n(n\plus{}3)$.
1998 Mexico National Olympiad, 5
The tangents at points $B$ and $C$ on a given circle meet at point $A$. Let $Q$ be a point on segment $AC$ and let $BQ$ meet the circle again at $P$. The line through $Q $ parallel to $AB$ intersects $BC$ at $J$. Prove that $PJ$ is parallel to $AC$ if and only if $BC^2 = AC\cdot QC$.
2017 USA TSTST, 3
Consider solutions to the equation \[x^2-cx+1 = \dfrac{f(x)}{g(x)},\] where $f$ and $g$ are polynomials with nonnegative real coefficients. For each $c>0$, determine the minimum possible degree of $f$, or show that no such $f,g$ exist.
[i]Proposed by Linus Hamilton and Calvin Deng[/i]
2023 Romania National Olympiad, 1
We consider real positive numbers $a,b,c$ such that $a + b + c = 3.$
Prove that $a^2 + b^2 + c^2 + a^2b + b^2 c + c^2 a \ge 6.$
2015 AMC 12/AHSME, 10
How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
2004 BAMO, 4
Suppose one is given $n$ real numbers, not all zero, but such that their sum is zero.
Prove that one can label these numbers $a_1, a_2, ..., a_n$ in such a manner that $a_1a_2 + a_2a_3 +...+a_{n-1}a_n + a_na_1 < 0$.