Olympiad problems

2014 Miklós Schweitzer, 1

Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

2002 Greece Junior Math Olympiad, 4

Tags: inequalities , induction , function

Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$

2016 Dutch IMO TST, 4

Tags: geometry , circles , fixed

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.

2024 pOMA, 4

Tags: geometry

Let $ABC$ be a triangle, and let $D$ and $E$ be two points on side $BC$ such that $BD = EC$. Let $X$ be a point on segment $AD$ such that $CX$ is parallel to the bisector of $\angle ADB$. Similarly, let $Y$ be a point on segment $AD$ such that $BY$ is parallel to the bisector of $\angle ADC$. Prove that $DE = XY$.

1990 ITAMO, 2

Tags: geometry , incenter , circumcircle

In a triangle $ABC$, the bisectors of the angles at $B$ and $A$ meet the opposite sides at $P$ and $Q$, respectively. Suppose that the circumcircle of triangle $PQC$ passes through the incenter $R $ of $\vartriangle ABC$. Given that $PQ = l$, find all sides of triangle $PQR$.

2013 BMT Spring, 6

Tags: combinatorics

In a class of $30$ students, each students knows exactly six other students. (Of course, knowing is a mutual relation, so if $A$ knows $B$, then $B$ knows $A$). A group of three students is balanced if either all three students know each other, or no one knows anyone else within that group. How many balanced groups exist?

2023 Swedish Mathematical Competition, 4

Tags: algebra , combinatorics

Let $f$ be a function that associates a positive integer $(x, y)$ with each pair of positive integers $f(x, y)$. Suppose that $f(x, y) \le xy$ for all positive integers $x$, $y$. Show that there are $2023$ different pairs $(x_1, y_1)$,$...$, $ (x_{2023}, y_{2023}$) such that $$f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}).$$

2006 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities , algebra

Prove that $\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ba} \ge a + b + c$, for all positive real numbers $a, b$, and $c$.

2024 Nigerian MO Round 3, Problem 3

Tags: geometry , circumcircle , trigonometry

Let $ABC$ be a triangle, and let $O$ be its circumcenter. Let $\overline{CO}\cap AB\equiv D$. Let $\angle BAC=\alpha$, and $\angle CBA=\beta$. Prove that $$\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|$$\\ For clarification, $\overline{CO}$ represents the line $CO$, and $AC$ represents the segment $AC$. Cases in which $D$ doesn't exist should be ignored.

1949-56 Chisinau City MO, 7

Tags: factorial , number theory , divisible , prime , divide

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.