Olympiad problems

1996 Turkey Team Selection Test, 1

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

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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

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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 $f(a_0), f(a_1), ...$.

2005 Taiwan National Olympiad, 1

Tags: number theory

Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.

2002 Mexico National Olympiad, 3

Tags: modular arithmetic , induction , strong induction , number theory , bijection

Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?

2024 LMT Fall, 14

Tags: team

Let $ABCD$ be an isosceles trapezoid with $2DA=2AB=2BC=CD$. A point $P$ lies in the interior of $ABCD$ such that $BP=1$, $CP=2$, $DP=4$. Find the area of $ABCD$.

2017 Brazil Undergrad MO, 6

Tags: combinatorics of words , combinatorics

Let's consider words over the alphabet $\{a,b\}$ to be sequences of $a$ and $b$ with finite length. We say $u \leq v$ if $u$ is a subword of $v$ if we can get $u$ erasing some letter of $v$ (for example $aba \leq abbab$). We say that $u$ differentiates the words $x$ and $y$ if $u \leq x$ but $u \not\leq y$ or vice versa. Let $m$ and $l$ be positive integers. We say that two words are $m-$equivalents if there does not exist some $u$ with length smaller than $m$ that differentiates $x$ and $y$. a) Show that, if $2m \leq l$, there exists two distinct words with length $l$ \ $m-$equivalents. b) Show that, if $2m > l$, any two distinct words with length $l$ aren't $m-$equivalent.

2018 Danube Mathematical Competition, 4

Tags: combinatorics

Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$. \\ \\ Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.

2011 Morocco National Olympiad, 3

Tags: function , search , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

2010 USAMO, 1

Tags: geometry , rectangle

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

1970 IMO Longlists, 59

Tags: modular arithmetic , digit , equation , number theory

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

2016 BMT Spring, 4

Tags: algebra

An geometric progression starting at $a_0 = 3$ has an even number of terms. Suppose the difference between the odd indexed terms and even indexed terms is $39321$ and that the sum of the first and last term is $49155$. Find the common ratio of this geometric progression.

2014 Saudi Arabia GMO TST, 1

Tags: geometry , angle bisector , circles

Let $A, B,C$ be colinear points in this order, $\omega$ an arbitrary circle passing through $B$ and $C$, and $l$ an arbitrary line different from $BC$, passing through A and intersecting $\omega$ at $M$ and $N$. The bisectors of the angles $\angle CMB$ and $\angle CNB$ intersect $BC$ at $P$ and $Q$. Prove that $AP\cdot AQ = AB \cdot AC$.

2003 Moldova Team Selection Test, 3

Tags: geometry

Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$