Olympiad problems

2011 QEDMO 9th, 3

A numerist has $n$ eurodollars and distributes them to two bank accounts $A, B$ in Germany and Switzerland, whereby the Eurodollars cannot be split into smaller monetary units due to the lack of a suitable name. In order to hide all money from the tax authorities if necessary, he would like to be able to move all of his money to account $B$. Due to the immense bureaucracy, money is only allowed to be moved between two accounts if the deposited amount in one account is double. Of course, he can carry out several such transfers in a row. Show that the number of ways to initially distribute the money appropriately is a power of two.

2023 Purple Comet Problems, 13

Tags: geometry

In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$.

2008 Indonesia TST, 4

Tags: algebra , inequalities

Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$

2017 All-Russian Olympiad, 1

Tags: number theory , trigonometry

$S=\sin{64x}+\sin{65x}$ and $C=\cos{64x}+\cos{65x}$ are both rational for some $x$. Prove, that for one of these sums both summands are rational too.

2008 Bulgarian Autumn Math Competition, Problem 9.2

Tags: geometry , circumcircle , touching circles , ratio

Given a $\triangle ABC$ and the altitude $CH$ ($H$ lies on the segment $AB$) and let $M$ be the midpoint of $AC$. Prove that if the circumcircle of $\triangle ABC$, $k$ and the circumcircle of $\triangle MHC$, $k_{1}$ touch, then the radius of $k$ is twice the radius of $k_{1}$.

1985 IMO Longlists, 76

Tags: quadratic , number theory

Are there integers $m$ and $n$ such that \[5m^2 - 6mn + 7n^2 = 1985 \ ?\]

2020 Thailand TST, 6

Tags: geometry

Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$. (Slovakia)

1977 All Soviet Union Mathematical Olympiad, 247

Tags: square table , combinatorics , combinatorial geometry

Given a square $100\times 100$ on the sheet of cross-lined paper. There are several broken lines drawn inside the square. Their links consist of the small squares sides. They are neither pairwise- nor self-intersecting (have no common points). Their ends are on the big square boarder, and all the other vertices are in the big square interior. Prove that there exists (in addition to four big square angles) a node (corresponding to the cross-lining family, inside the big square or on its side) that does not belong to any broken line.

1996 Turkey Team Selection Test, 1

Tags: ratio , parallelogram , geometry

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $S_{ABC} = S_{ADC}$ intersect at $E$. The lines through $E$ parallel to $AD$, $DC$, $CB$, $BA$ meet $AB$, $BC$, $CD$, $DA$ at $K$, $L$, $M$, $N$, respectively. Compute the ratio $\frac{S_{KLMN}}{S_{ABC}}$

2010 AMC 10, 6

Tags:

A circle is centered at $ O$, $ \overline{AB}$ is a diameter and $ C$ is a point on the circle with $ \angle COB \equal{} 50^{\circ}$. What is the degree measure of $ \angle CAB$? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 65$

1978 AMC 12/AHSME, 24

Tags: ratio , geometric sequence

If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation $\textbf{(A) }r^2+r+1=0\qquad\textbf{(B) }r^2-r+1=0\qquad\textbf{(C) }r^4+r^2-1=0$ $\qquad\textbf{(D) }(r+1)^4+r=0\qquad \textbf{(E) }(r-1)^4+r=0$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.8

Tags: algebra , inequalities , trigonometry

Find all $x$ for which the inequality holds $$\sqrt{7+8x-16x^2} \ge 2^{\cos^2 \pi x}+2^{\sin ^2 \pi x}$$

2011 Singapore MO Open, 5

Tags: modular arithmetic , algebra , function , domain , number theory

Find all pairs of positive integers $(m,n)$ such that \[m+n-\frac{3mn}{m+n}=\frac{2011}{3}.\]

2006 Belarusian National Olympiad, 8

Tags: number theory , perfect square

a) Do there exist positive integers $a$ and $b$ such that for any positive,integer $n$ the number $a \cdot 2^n+ b\cdot 5^n$ is a perfect square ? b) Do there exist positive integers $a, b$ and $c$, such that for any positive integer $n$ the number $a\cdot 2^n+ b\cdot 5^n + c$ is a perfect square? (M . Blotski)

2009 Today's Calculation Of Integral, 496

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Evaluate $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$ You may not use $ \tan ^{ \minus{} 1} x$ or Complex Integral here.

2021/2022 Tournament of Towns, P4

Tags:

A rock travelled through an n x n board, stepping at each turn to the cell neighbouring the previous one by a side, so that each cell was visited once. Bob has put the integer numbers from 1 to n^2 into the cells, corresponding to the order in which the rook has passed them. Let M be the greatest difference of the numbers in neighbouring by side cells. What is the minimal possible value of M?

1998 APMO, 3

Tags: inequalities , trigonometry , algebra

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr). \]

1999 Mongolian Mathematical Olympiad, Problem 3

Tags: number theory , sequence

Does there exist a sequence $(a_n)_{n\in\mathbb N}$ of distinct positive integers such that: (i) $a_n<1999n$ for all $n$; (ii) none of the $a_n$ contains three decimal digits $1$?

1975 Bundeswettbewerb Mathematik, 2

Tags: number theory , prime number

Prove that no term of the sequence $10001$, $100010001$, $1000100010001$ , $...$ is prime.

2013 Brazil Team Selection Test, 4

Tags: algebra , polynomial , combinatorics

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

2011 Postal Coaching, 1

Tags: geometry , incenter , geometric transformation , circumcircle , reflection , inradius

Let $I$ be the incentre of a triangle $ABC$ and $\Gamma_a$ be the excircle opposite $A$ touching $BC$ at $D$. If $ID$ meets $\Gamma_a$ again at $S$, prove that $DS$ bisects $\angle BSC$.

1965 Swedish Mathematical Competition, 4

Tags: function , recurrence relation , sequence , algebra

Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.