Olympiad problems

2011 Serbia National Math Olympiad, 1

Let $n \ge 2$ be integer. Let $a_0$, $a_1$, ... $a_n$ be sequence of positive reals such that: $(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$, for $k=1, 2, ..., n-1$. Prove $a_n< \frac{1}{n-1}$.

1989 All Soviet Union Mathematical Olympiad, 508

Tags: polyhedron , combinatorial geometry , combinatorics , 3d geometry , even

A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).

1996 Romania National Olympiad, 3

Tags: trigonometry , invariant , algebra , identity

Prove that $ \forall x\in \mathbb{R} $ , $ \cos ^7x+\cos ^7(x+\frac {2\pi}{3})+\cos ^7(x+\frac {4\pi}{3})=\frac {63}{64}\cos 3x $

2010 AMC 12/AHSME, 1

Tags: percent

Makayla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$

1992 Austrian-Polish Competition, 7

Tags: 3d geometry , solid geometry , geometric inequality , right angle , geometry

Consider triangles $ABC$ in space. (a) What condition must the angles $\angle A, \angle B , \angle C$ of $\triangle ABC$ fulfill in order that there is a point $P$ in space such that $\angle APB, \angle BPC, \angle CPA$ are right angles? (b) Let $d$ be the longest of the edges $PA,PB,PC$ and let $h$ be the longest altitude of $\triangle ABC$. Show that $\frac{1}{3}\sqrt6 h \le d \le h$.

2006 Italy TST, 3

Tags: function , induction , algebra , functional equation

Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$, \[f(m - n + f(n)) = f(m) + f(n).\]

Durer Math Competition CD Finals - geometry, 2019.C5

Tags: geometry , equilateral , distance

$A, B, C, D$ are four distinct points such that triangles $ABC$ and $CBD$ are both equilateral. Find as many circles as you can, which are equidistant from the four points. How can these circles be constructed? [i]Remark: The distance between a point $P$ and a circle c is measured as follows: we join $P$ and the centre of the circle with a straight line, and measure how much we need to travel along thisline (starting from $P$) to hit the perimeter of the circle. If $P$ is an internal point of the circle, the distance is the length of the shorter such segment. The distance between a circle and itscentre is the radius of the circle.[/i]

2005 APMO, 5

Tags: circumcircle , inradius , ratio , incenter , euler , trigonometry , geometry

In a triangle $ABC$, points $M$ and $N$ are on sides $AB$ and $AC$, respectively, such that $MB = BC = CN$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $ABC$, respectively. Express the ratio $MN/BC$ in terms of $R$ and $r$.

1940 Moscow Mathematical Olympiad, 059

Tags: number theory , integer sequence , digit

Consider all positive integers written in a row: $123456789101112131415...$ Find the $206788$-th digit from the left.

1977 IMO Longlists, 11

Tags: pigeonhole principle , modular arithmetic , number theory

Let $n$ and $z$ be integers greater than $1$ and $(n,z)=1$. Prove: (a) At least one of the numbers $z_i=1+z+z^2+\cdots +z^i,\ i=0,1,\ldots ,n-1,$ is divisible by $n$. (b) If $(z-1,n)=1$, then at least one of the numbers $z_i$ is divisible by $n$.

1986 AMC 12/AHSME, 17

Tags: pigeonhole principle

A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.) $ \textbf{(A)}\ 21\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 50$

1955 AMC 12/AHSME, 23

Tags:

In checking the petty cash a clerk counts $ q$ quarters, $ d$ dimes, $ n$ nickels, and $ c$ cents. Later he discovers that $ x$ of the nickels were counted as quarters and $ x$ of the dimes were counted as cents. To correct the total obtained the clerk must: $ \textbf{(A)}\ \text{make no correction} \qquad \textbf{(B)}\ \text{subtract 11 cents} \qquad \textbf{(C)}\ \text{subtract 11}x\text{ cents} \\ \textbf{(D)}\ \text{add 11}x\text{ cents} \qquad \textbf{(E)}\ \text{add }x\text{ cents}$

2014 AMC 12/AHSME, 12

Tags: ratio , geometry , trigonometry , trig identities , law of cosines

Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle? $\textbf{(A) }2\qquad \textbf{(B) }1+\sqrt3\qquad \textbf{(C) }3\qquad \textbf{(D) }2+\sqrt3\qquad \textbf{(E) }4\qquad$

2021-2022 OMMC, 21

Tags:

For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \tfrac pq$ for coprime positive integers $p$, $q$, find $p + q$. [i]Proposed by Justin Lee[/i]

2007 Korea National Olympiad, 1

Tags: inequalities

For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality? $ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ .

1952 Moscow Mathematical Olympiad, 227

Tags: combinatorial geometry , plane , lines in plane

$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.

1979 Dutch Mathematical Olympiad, 3

Tags: recurrence relation , sequence , number theory , last digit

Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.

2013 NIMO Problems, 8

Tags:

Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$. [i]Based on a proposal by Ivan Koswara[/i]

2008 Sharygin Geometry Olympiad, 6

Tags: geometric transformation , reflection , circumcircle , geometry

(B.Frenkin) Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.

2010 Contests, 2

Tags: ratio , geometry

Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$.

2023/2024 Tournament of Towns, 7

Tags: combinatorics

7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , max , area , midpoint , sidelengths

Let $ABC$ be a triangle and $a = BC, b = CA$ and $c = AB$ be the lengths of its sides. Points $D$ and $E$ lie in the same halfplane determined by $BC$ as $A$. Suppose that $DB = c, CE = b$ and that the area of $DECB$ is maximal. Let $F$ be the midpoint of $DE$ and let $FB = x$. Prove that $FC = x$ and $4x^3 = (a^2+b^2 + c^2)x + abc$.

2010 Contests, 3

Tags: geometry , parallelogram , circumcircle , trigonometry , trapezoid , trig identities , law of sines

$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.

1999 Spain Mathematical Olympiad, 6

Tags: combinatorial geometry , parallel lines , minimum , plane , combinatorics

A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?

1983 Bulgaria National Olympiad, Problem 2

Tags: inequalities

Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$, $$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$