Olympiad problems

V Soros Olympiad 1998 - 99 (Russia), 9.5

Tags: geometry , area

In triangle $ABC$, $\angle BAC= 60^o$. Point $P$ is taken inside the triangle so that angles $\angle APB=\angle BPC= \angle CP A=120^o$. It is known that $AP = a$. Find the area of triangle $BPC$.

2015 ASDAN Math Tournament, 17

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How many ways are there to write $91$ as the sum of at least $2$ consecutive positive integers?

2008 Romania National Olympiad, 1

Tags: trigonometry , geometry , circumcircle , geometric transformation

Let $ ABC$ be an acute angled triangle with $ \angle B > \angle C$. Let $ D$ be the foot of the altitude from $ A$ on $ BC$, and let $ E$ be the foot of the perpendicular from $ D$ on $ AC$. Let $ F$ be a point on the segment $ (DE)$. Show that the lines $ AF$ and $ BF$ are perpendicular if and only if $ EF\cdot DC \equal{} BD \cdot DE$.

2010 Math Prize For Girls Problems, 3

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How many ordered triples of integers $(x, y, z)$ are there such that \[ x^2 + y^2 + z^2 = 34 \, ? \]

2007 AMC 12/AHSME, 17

Tags: trigonometry , function

Suppose that $ \sin a \plus{} \sin b \equal{} \sqrt {\frac {5}{3}}$ and $ \cos a \plus{} \cos b \equal{} 1.$ What is $ \cos(a \minus{} b)?$ $ \textbf{(A)}\ \sqrt {\frac {5}{3}} \minus{} 1 \qquad \textbf{(B)}\ \frac {1}{3}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {2}{3}\qquad \textbf{(E)}\ 1$

2006 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: combinatorics

The numbers $1, 2,\ldots, 2006$ are written around the circumference of a circle. A [i]move[/i] consists of exchanging two adjacent numbers. After a sequence of such moves, each number ends up $13$ positions to the right of its initial position. lf the numbers $1, 2,\ldots, 2006$ are partitioned into $1003$ distinct pairs, then show that in at least one of the moves, the two numbers of one of the pairs were exchanged.

2009 Brazil Team Selection Test, 1

Tags: right triangle , orthocenter , geometry , cyclic , pentagon

Let $A, B, C, D, E$ points in circle of radius r, in that order, such that $AC = BD = CE = r$. The points $H_1, H_2, H_3$ are the orthocenters of the triangles $ACD$, $BCD$ and $BCE$, respectively. Prove that $H_1H_2H_3$ is a right triangle .

2003 Croatia National Olympiad, Problem 3

Tags: inequalities

For positive numbers $a_1,a_2,\ldots,a_n$ ($n\ge2$) denote $s=a_1+\ldots+a_n$. Prove that $$\frac{a_1}{s-a_1}+\ldots+\frac{a_n}{s-a_n}\ge\frac n{n-1}.$$

2021 German National Olympiad, 4

Tags: length , similar triangles , parallelogram , geometry

Let $OFT$ and $NOT$ be two similar triangles (with the same orientation) and let $FANO$ be a parallelogram. Show that \[\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.\]

MIPT student olimpiad autumn 2022, 4

Tags: geometry

In $R^n$ space is given a finite set of points $X$. It is known that for any subset $Y \subseteq X$ of at most $n+1$ points, there is a unit ball $B_Y$ containing $Y$ and not containing the origin. Prove that there is a unit a ball $B_X$ containing $X$ and not containing the origin.

1968 IMO Shortlist, 26

Tags: function , algebra , periodic function , functional equation

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

2005 MOP Homework, 5

Tags: number theory

Find all integer solutions to $y^2(x^2+y^2-2xy-x-y)=(x+y)^2(x-y)$.

2004 Purple Comet Problems, 11

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Find the sum of all integers $x$ satisfying $1 + 8x \le 358 - 2x \le 6x + 94$.

1987 IMO Longlists, 32

Tags: number theory , inequalities

Solve the equation $28^x = 19^y +87^z$, where $x, y, z$ are integers.

1998 South africa National Olympiad, 5

Tags: greatest common divisor , number theory

Prove that \[ \gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)} \] is a prime if $n$ is a power of a prime, and 1 otherwise.

2018 Korea National Olympiad, 4

Tags: algebra , inequality , inequalities , sequence

Find all real values of $K$ which satisfies the following. Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$. (i). $0 < a_n < n^K$. (ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$. Then, there exists a positive integer $N$ such that for all integers $n>N$, $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$

2006 QEDMO 2nd, 12

Tags: induction , additive number theory

Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.

1966 All Russian Mathematical Olympiad, 078

Tags: convex polygon , circles , geometry

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

2014 Albania Round 2, 3

Tags: geometry

In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$

2024 Austrian MO Regional Competition, 1

Tags: inequalities , algebra

Let $a$, $b$ and $c$ be real numbers larger than $1$. Prove the inequality $$\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.$$ When does equality hold? [i](Karl Czakler)[/i]

2021 Junior Macedonian Mathematical Olympiad, Problem 4

Tags: algebra , inequality

Let $a$, $b$, $c$ be positive real numbers such that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} = \frac{27}{4}.$ Show that: $$\frac{a^3+b^2}{a^2+b^2} + \frac{b^3+c^2}{b^2+c^2} + \frac{c^3+a^2}{c^2+a^2} \geq \frac{5}{2}.$$ [i]Authored by Nikola Velov[/i]

2023 Bulgaria National Olympiad, 6

Tags: combinatorics

In a class of $26$ students, everyone is being graded on five subjects with one of three possible marks. Prove that if $25$ of these students have received their marks, then we can grade the last one in such a way that their marks differ from these of any other student on at least two subjects.

2010 Math Prize For Girls Problems, 4

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Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$. The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$? (The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.)

2025 India STEMS Category A, 6

Tags: polynomial , algebra

Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots. (Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$). [i]Proposed by Malay Mahajan[/i]

2002 AIME Problems, 1

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Given that \begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& \text{y is the number formed by reversing the digits of x; and}\\ &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of $z$ are possible?