Olympiad problems

2010 Morocco TST, 3

Tags: function , algebra

Let $G$ be a non-empty set of non-constant functions $f$ such that $f(x)=ax + b$ (where $a$ and $b$ are two reals) and satisfying the following conditions: 1) if $f \in G$ and $g \in G$ then $gof \in G$, 2) if $f \in G$ then $f^ {-1} \in G$, 3) for all $f \in G$ there exists $x_f \in \mathbb{R}$ such that $f(x_f)=x_f$. Prove that there is a real $k$ such that for all $f \in G$ we have $f(k)=k$

2021-IMOC, A8

Tags: functional equation , algebra , inequality , function , inequalities

Find all functions $f : \mathbb{N} \to \mathbb{N}$ with $$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$ holds for all positive integers $x,y.$

2014 239 Open Mathematical Olympiad, 7

Tags: geometry

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

2023 Novosibirsk Oral Olympiad in Geometry, 6

Tags: geometry , equal segments , isosceles , angle

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

1930 Eotvos Mathematical Competition, 2

Tags: combinatorics , combinatorial geometry , chessboard , geometry

A straight line is drawn across an $8\times 8$ chessboard. It is said to [i]pierce [/i]a square if it passes through an interior point of the square. At most how many of the $64$ squares can this line [i]pierce[/i]?

1967 IMO Longlists, 9

Tags: incenter , angle bisector , locus , locus problems , geometry

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

2024 HMNT, 2

Tags: guts

Compute the smallest integer $n > 72$ that has the same set of prime divisors as $72.$

1990 IMO Longlists, 83

Tags: geometry

Point $D$ is on the hypotenuse $BC$ of right-angled triangle $ABC$. The inradii of triangles $ADB$ and $ADC$ are equal. Prove that $S_{ABC} = AD^2$, where $S$ is the area function.

2008 National Olympiad First Round, 26

Tags: factorial

Let $A=\frac{2^2+3\cdot 2 + 1}{3! \cdot 4!} + \frac{3^2+3\cdot 3 + 1}{4! \cdot 5!} + \frac{4^2+3\cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2+3\cdot 10 + 1}{11! \cdot 12!}$. What is the remainder when $11!\cdot 12! \cdot A$ is divided by $11$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10 $

2013 ELMO Problems, 4

Tags: geometry , incenter , angle bisector , perpendicular bisector

Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$. [i]Proposed by Evan Chen[/i]