Olympiad problems

2013 Oral Moscow Geometry Olympiad, 4

Tags: geometry , circumcircle , angle bisector , equal segments , concurrency , concurrent

Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.

2014 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry

Incircle $\omega$ of $ABC$ touch $AC$ at $B_1$. Point $E,F$ on the $\omega$ such that $\angle AEB_1=\angle B_1FC=90$. Tangents to $\omega$ at $E,F$ intersects in $D$, and $B$ and $D$ are on different sides for line $AC$. $M$- midpoint of $AC$. Prove, that $AE,CF,DM$ intersects at one point.

1970 IMO Longlists, 34

Tags:

In connection with a convex pentagon $ABCDE$ we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.

2022 Brazil National Olympiad, 3

Tags: geometry

Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.

2005 iTest, 31

Tags: number theory , Digits

Let $X = 123456789$. Find the sum of the tens digits of all integral multiples of $11$ that can be obtained by interchanging two digits of $X$.

2021 CHMMC Winter (2021-22), 5

Tags: algebra

Find all functions $f : R \to R$ such that $$f(f(x) + f(y)^2) = f(x)^2 +y^2f(y)^3.$$ Here $R$ denotes the usual real numbers.

2012 Kazakhstan National Olympiad, 1

Tags: algebra , difference of squares , special factorizations , number theory unsolved , number theory , Kazakhstan

Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.

Russian TST 2018, P2

Tags: combinatorics , set theory , game , TST , Russian TST , Game Theory

Let $\mathcal{F}$ be a finite family of subsets of some set $X{}$. It is known that for any two elements $x,y\in X$ there exists a permutation $\pi$ of the set $X$ such that $\pi(x)=y$, and for any $A\in\mathcal{F}$ \[\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.\]A bear and crocodile play a game. At a move, a player paints one or more elements of the set $X$ in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in $\mathcal{F}$ in his own color loses. If this does not happen and all the elements of $X$ have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.

2014 Chile TST IMO, 2

Tags: Chile , number theory , LTE Lemma

Given $n, k \in \mathbb{N}$, prove that $(n-1)^2$ divides $n^k - 1$ if and only if $n-1 \mid k$.

2006 Iran Team Selection Test, 4

Tags: inequalities , induction , function , triangle inequality , inequalities proposed

Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that \[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]

Cono Sur Shortlist - geometry, 2005.G4.2

Tags: geometry , circumcircle , trigonometry , geometry proposed

Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively. (a) Show that $R$, $Q$, $W$, $S$ are collinear. (b) Show that $MP=RS-QW$.

2008 Abels Math Contest (Norwegian MO) Final, 1

Tags: number theory , divisible , Integer

Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$. (a) Show that $s(n)$ is an integer whenever $n$ is an integer. (b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?

2013 Hanoi Open Mathematics Competitions, 3

Tags: floor function , algebra

The largest integer not exceeding $[(n+1)a]-[na]$ where $n$ is a natural number, $a=\frac{\sqrt{2013}}{\sqrt{2014}}$ is: (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

2004 Oral Moscow Geometry Olympiad, 1

Tags: areas , geometry

In a convex quadrilateral $ABCD$, $E$ is the midpoint of $CD$, $F$ is midpoint of $AD$, $K$ is the intersection point of $AC$ with $BE$. Prove that the area of triangle $BKF$ is half the area of triangle $ABC$.

PEN J Problems, 4

Tags: algebra , polynomial , arithmetic sequence , Divisor Functions

Let $m$, $n$ be positive integers. Prove that, for some positive integer $a$, each of $\phi(a)$, $\phi(a+1)$, $\cdots$, $\phi(a+n)$ is a multiple of $m$.

2021 Belarusian National Olympiad, 8.4

Tags: combinatorics

Several soldiers are standing in a row. After a command each of them turned their head either to the left or to the right. After that every second every soldier performs the following operation simultaneously: 1) if the soldier is facing right and the majority of soldiers to the right of him are facing left, he starts facing left; 2) if the soldier is facing left and the majority of soldiers to the left of him are facing right, he starts facing right; 3) otherwise he does nothing. Prove that at some point the process will stop.

2024 USA IMO Team Selection Test, 5

Tags: USA TST 2024 , USA TST , number theory

Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences. [i]Ray Li[/i]

2024 CMIMC Geometry, 1

Tags: geometry

Let $ABCD$ be a rectangle with $AB=5$. Let $E$ be on $\overline{AB}$ and $F$ be on $\overline {CD}$ such that $AE=CF=4$. Let $P$ and $Q$ lie inside $ABCD$ such that triangles $AEP$ and $CFQ$ are equilateral. If $E$, $P$, $Q$, and $F$ lie on a single line, find $\overline{BC}$. [i]Proposed by Connor Gordon[/i]

2013 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , angle bisector , circumcircle , equal segments , altitudes

In the acute-angled triangle $ABC$, with $AB \ne AC$, $D$ is the foot of the angle bisector of angle $A$, and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. The circumcircles of triangles $DBF$ and $DCE$ intersect for the second time at $M$. Prove that $ME = MF$. Leonard Giugiuc

2020 Austrian Junior Regional Competition, 3

Tags: geometry , trapezoid

Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$. (Karl Czakler)

2020 USMCA, 14

Tags:

Call a real number [i]amiable[/i] if it can be expressed in the form $a - b\sqrt{2}$, where $1 \le a, b \le 100$ are integers. Find the amiable number $x$ that minimizes $\left|x - \frac{1}{3}\right|$.

1991 Romania Team Selection Test, 3

Tags: inequalities , Coloring , combinatorics , polygon

Let $C$ be a coloring of all edges and diagonals of a convex $n$−gon in red and blue (in Romanian, rosu and albastru). Denote by $q_r(C)$ (resp. $q_a(C)$) the number of quadrilaterals having all its edges and diagonals red (resp. blue). Prove: $ \underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}$

2008 Nordic, 1

Tags: function , algebra unsolved , algebra

Find all reals $A,B,C$ such that there exists a real function $f$ satisfying $f(x+f(y))= Ax+By+C$ for all reals $x,y$.

2008 Harvard-MIT Mathematics Tournament, 1

Tags: Pythagorean Theorem , geometry

Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$, and also that $ AX \equal{} \frac {\sqrt {2}}{2}$. Determine the value of $ CX^2$.

2001 China Team Selection Test, 1

Tags: geometry , China , China TST

In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.