Olympiad problems

2011 IMAR Test, 3

Given an integer number $n \ge 2$, show that there exists a function $f : R \to R$ such that $f(x) + f(2x) + ...+ f(nx) = 0$, for all $x \in R$, and $f(x) = 0$ if and only if $x = 0$.

1991 Bundeswettbewerb Mathematik, 2

Tags: induction , graph theory , combinatorics

In the space there are 8 points that no four of them are in the plane. 17 of the connecting segments are coloured blue and the other segments are to be coloured red. Prove that this colouring will create at least four triangles. Prove also that four cannot be subsituted by five. Remark: Blue triangles are those triangles whose three edges are coloured blue.

Kyiv City MO Juniors Round2 2010+ geometry, 2010.89.3

Tags: geometry , incenter , angle

In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles $ABH ,CBH$ respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$.

2014 Contests, 4

Tags: geometry

For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$, let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$.Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$, respectively. Likewise, let $S$ and $T$ be the intersections of the parallel to $BC$ through $P$ with the sides $AB$ and $AC$, respectively.In a given triangle $ABC$, determine all points $P$ for which the triangles $PRD$, $PEQ$and $PTE$ have the same area.

2001 All-Russian Olympiad Regional Round, 10.4

Tags: geometry , combinatorics , combinatorial geometry

Three families of parallel lines are drawn,$10$ lines each, are drawn. What is the greatest number of triangles they can cut from plane?

2022 IMO Shortlist, N5

Tags: number theory

For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

1970 IMO Longlists, 20

Tags: geometry , 3d geometry , tetrahedron , volume , equation

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

1968 AMC 12/AHSME, 24

Tags: geometry , ratio

A painting $18''\ \text{X}\ 24''$ is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is: $\textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:2 \qquad\textbf{(C)}\ 2:3 \qquad\textbf{(D)}\ 3:4 \qquad\textbf{(E)}\ 1:1$

III Soros Olympiad 1996 - 97 (Russia), 9.1

Tags: number theory

In the middle of the number 1996 (i.e. between $19$ and $96$) you need to insert several digits so that the resulting number is divisible by $1997$. In this case, you need to get by with the least number of inserted digits.

2006 Estonia National Olympiad, 1

Tags: combinatorics

We call a [i]ship[/i] a figure made up of unit squares connected by common edges. Prove that if there is an odd number of possible different ships consisting of n unit squares on a $ 10 \times 10$ board, then n is divisible by 4.