Found problems: 85335
2012 Turkmenistan National Math Olympiad, 5
Let $O$ be the center of $\bigtriangleup ABC$'s circumcircle. $CO$ line intersect $AB$ at $D$ and $BO$ line intersect $AC$ at $E$. If $\angle A=\angle CDE=50$° then find $\angle ADE$
2014 ASDAN Math Tournament, 9
A sequence $\{a_n\}_{n\geq0}$ obeys the recurrence $a_n=1+a_{n-1}+\alpha a_{n-2}$ for all $n\geq2$ and for some $\alpha>0$. Given that $a_0=1$ and $a_1=2$, compute the value of $\alpha$ for which
$$\sum_{n=0}^{\infty}\frac{a_n}{2^n}=10$$
2015 IMC, 5
Let $n\ge2$, let $A_1,A_2,\ldots,A_{n+1}$ be $n+1$ points in the
$n$-dimensional Euclidean space, not lying on the same hyperplane,
and let $B$ be a point strictly inside the convex hull of
$A_1,A_2,\ldots,A_{n+1}$. Prove that $\angle A_iBA_j>90^\circ$ holds
for at least $n$ pairs $(i,j)$ with $\displaystyle{1\le i<j\le
n+1}$.
Proposed by Géza Kós, Eötvös University, Budapest
1994 Korea National Olympiad, Problem 3
Let $\alpha,\beta ,\gamma$ be the angles of $\triangle ABC$.
a) Show that $cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma$ .
b) Given that $cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25$, find $sin\alpha : sin\beta : sin\gamma$ .
2025 Kosovo National Mathematical Olympiad`, P3
Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?
1988 USAMO, 2
The cubic equation $x^3 + ax^2 + bx + c = 0$ has three real roots. Show that $a^2-3b\geq 0$, and that $\sqrt{a^2-3b}$ is less than or equal to the difference between the largest and smallest roots.
2021 Switzerland - Final Round, 5
For which integers $n \ge 2$ can we arrange numbers $1,2, \ldots, n$ in a row, such that for all integers $1 \le k \le n$ the sum of the first $k$ numbers in the row is divisible by $k$?
2011 Postal Coaching, 4
Let $a, b, c$ be positive integers for which \[ac = b^2 + b + 1\] Prove that the equation
\[ax^2 - (2b + 1)xy + cy^2 = 1\]
has an integer solution.
2012 Albania Team Selection Test, 2
It is given an acute triangle $ABC$ , $AB \neq AC$ where the feet of altitude from $A$ its $H$. In the extensions of the sides $AB$ and $AC$ (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.
Find the ratio $\tfrac{HP}{HA}$.
2022 Purple Comet Problems, 12
Let $a$ and $b$ be positive real numbers satisfying
$$\frac{a}{b} \left( \frac{a}{b}+ 2 \right) + \frac{b}{a} \left( \frac{b}{a}+ 2 \right)= 2022.$$
Find the positive integer $n$ such that $$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=\sqrt{n}.$$
2005 Danube Mathematical Olympiad, 2
Prove that the sum:
\[ S_n=\binom{n}{1}+\binom{n}{3}\cdot 2005+\binom{n}{5}\cdot 2005^2+...=\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\binom{n}{2k+1}\cdot 2005^k \]
is divisible by $2^{n-1}$ for any positive integer $n$.
2006 MOP Homework, 7
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $T$. Let $X$ be a point on circle $\omega_1$. Line $l_1$ is tangent to circle $\omega_1$ and $X$, and line $l$ intersects circle $\omega_2$ at $A$ and $B$. Line $XT$ meets circle $\omega$ at $S$. Point $C$ lies on arc $TS$ (of circle $\omega_2$, not containing points $A$ and $B$). Point $Y$ lies on circle $\omega_1$ and line $YC$ is tangent to circle $\omega_1$. Let $I$ be the intersection of lines $XY$ ad $SC$. Prove that...
a) points $C$, $T$, $Y$, $I$ lie on a circle
(B) $I$ is an excenter of triangle $ABC$.
2019 IFYM, Sozopol, 1
A football tournament is played between 5 teams, each two of which playing exactly one match. 5 points are awarded for a victory and 0 – for a loss. In case of a draw 1 point is awarded to both teams, if no goals are scored, and 2 – if they have scored any. In the final ranking the five teams had points that were 5 consecutive numbers. Determine the least number of goals that could be scored in the tournament.
2019 Argentina National Olympiad, 5
There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression.
Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.
2012 HMNT, 7
Let $A_1A_2 . . .A_{100}$ be the vertices of a regular $100$-gon. Let $\pi$ be a randomly chosen permutation of the numbers from $1$ through $100$. The segments $A_{\pi (1)}A_{\pi (2)}$, $A_{\pi (2)}A_{\pi (3)}$, $...$ ,$A_{\pi (99)}A_{\pi (100)}, A_{\pi (100)}A_{\pi (1)}$ are drawn. Find the expected number of pairs of line segments that intersect at a point in the interior of the $100$-gon.
2022 Kosovo & Albania Mathematical Olympiad, 3
Is it possible to partition $\{1, 2, 3, \ldots, 28\}$ into two sets $A$ and $B$ such that both of the following conditions hold simultaneously:
(i) the number of odd integers in $A$ is equal to the number of odd integers in $B$;
(ii) the difference between the sum of squares of the integers in $A$ and the sum of squares of the integers in $B$ is $16$?
1971 Putnam, A3
The three vertices of a triangle of sides $a,b,$ and $c$ are lattice points and lie on a circle of radius $R$. Show that $abc \geq 2R.$ (Lattice points are points in Euclidean plane with integral coordinates.)
1995 Taiwan National Olympiad, 4
Let $m_{1},m_{2},...,m_{n}$ be mutually distinct integers. Prove that there exists a $f(x)\in\mathbb{Z}[x]$ of degree $n$ satisfying the following two conditions:
a)$f(m_{i})=-1\forall i=1,2,...,n$.
b)$f(x)$ is irreducible.
2018 Auckland Mathematical Olympiad, 5
Find all possible triples of positive integers, $a, b, c$ so that $\frac{a+1}{b}$, $\frac{b+1}{c}$ and $\frac{c+1}{a}$ are also integers.
1996 Romania National Olympiad, 3
Let $N, P$ be the centers of the faces A$BB'A'$ and $ADD'A'$, respectively, of a right parallelepiped $ABCDA'B'C'D'$ and $M \in (A'C)$ such that $A'M= \frac13 A' C$. Prove that $MN \perp AB'$ and $ MP \perp AD' $ if and only if the parallelepiped is a cube.
2010 ELMO Shortlist, 3
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that
\begin{align*}
a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a
\end{align*}
[i]Travis Hance.[/i]
2022 MOAA, 11
Let a [i]triplet [/i] be some set of three distinct pairwise parallel lines. $20$ triplets are drawn on a plane. Find the maximum number of regions these $60$ lines can divide the plane into.
1999 Akdeniz University MO, 3
For all $x> \sqrt 2$, $y> \sqrt 2$ numbers, prove that
$$x^4-x^3y+x^2y^2-xy^3+y^4>x^2+y^2$$
2003 Costa Rica - Final Round, 3
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.
2013 BMT Spring, 2
Two rays start from a common point and have an angle of $60$ degrees. Circle $C$ is drawn with radius $42$ such that it is tangent to the two rays. Find the radius of the circle that has radius smaller than circle $C$ and is also tangent to $C$ and the two rays.