Olympiad problems

1984 Brazil National Olympiad, 5

$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular.

2011 ELMO Shortlist, 2

Tags: coloring , directed graph , induction , graph theory , combinatorics

A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to. [i]David Yang.[/i] [hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]

1998 Hungary-Israel Binational, 3

Tags: combinatorics

Let $ n$ be a positive integer. We consider the set $ P$ of all partitions of $ n$ into a sum of positive integers (the order is irrelevant). For every partition $ \alpha$, let $ a_{k}(\alpha)$ be the number of summands in $ \alpha$ that are equal to $ k, k = 1,2,...,n.$ Prove that $ \sum_{\alpha\in P}\frac{1}{1^{a_{1}(\alpha)}a_{1}(\alpha)!\cdot 2^{a_{2}(\alpha)}a_{2}(\alpha)!...n^{a_{n}(\alpha)}a_{n}(\alpha)!}=1.$

Kvant 2022, M2719

Tags: residue , number theory

For an odd positive integer $n>1$ define $S_n$ to be the set of the residues of the powers of two, modulo $n{}$. Do there exist distinct $n{}$ and $m{}$ whose corresponding sets $S_n$ and $S_m$ coincide? [i]Proposed by D. Kuznetsov[/i]

2002 District Olympiad, 2

Tags: parallelogram , circumcircle , geometry

Let $ ABCD $ be an inscriptible quadrilateral and $ M $ be a point on its circumcircle, distinct from its vertices. Let $ H_1,H_2,H_3,H_4 $ be the orthocenters of $ MAB,MBC, MCD, $ respectively, $ MDA, $ and $ E,F, $ the midpoints of the segments $ AB, $ respectivley, $ CD. $ Prove that: [b]a)[/b] $ H_1H_2H_3H_4 $ is a parallelogram. [b]b)[/b] $ H_1H_3=2\cdot EF. $

2020 Stars of Mathematics, 1

Tags: inequality , algebra

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]

1998 Tuymaada Olympiad, 7

Tags: sequence , sum , arithmetic sequence , algebra , square

All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.

2020 Indonesia MO, 2

Tags: quadratic , algebra , polynomial

Problem 2. Let $P(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. If $$P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab$$ then prove that $$(a - b)(b - c)(c - a)(a + b + c) = 0.$$

1951 Miklós Schweitzer, 7

Tags: polynomial , algebra

Let $ f(x)$ be a polynomial with the following properties: (i) $ f(0)\equal{}0$; (ii) $ \frac{f(a)\minus{}f(b)}{a\minus{}b}$ is an integer for any two different integers $ a$ and $ b$. Is there a polynomial which has these properties, although not all of its coefficients are integers?

2008 Princeton University Math Competition, A1/B2

Tags: geometry

What is the area of a circle with a circumference of $8$?

1986 AMC 12/AHSME, 19

Tags: perimeter , analytic geometry , geometry

A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point? $ \textbf{(A)}\ \sqrt{13}\qquad\textbf{(B)}\ \sqrt{14}\qquad\textbf{(C)}\ \sqrt{15}\qquad\textbf{(D)}\ \sqrt{16}\qquad\textbf{(E)}\ \sqrt{17}$

2012 Cuba MO, 1

Tags: algebra , combinatorics

There are $1000$ balls of dough $0.38$ and $5000$ balls of dough $0.038$ that must be packed in boxes. A box contains a collection of balls whose total mass is at most $1$. Find the smallest number of boxes that they are needed.

1958 AMC 12/AHSME, 15

Tags:

A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is: $ \textbf{(A)}\ 1080\qquad \textbf{(B)}\ 900\qquad \textbf{(C)}\ 720\qquad \textbf{(D)}\ 540\qquad \textbf{(E)}\ 360$

1991 India National Olympiad, 8

Tags:

There are $10$ objects of total weight $20$, each of the weights being a positive integers. Given that none of the weights exceeds $10$ , prove that the ten objects can be divided into two groups that balance each other when placed on 2 pans of a balance.

2022 Cyprus JBMO TST, 3

Tags: inequalities , algebra

If $x,y$ are real numbers with $x+y\geqslant 0$, determine the minimum value of the expression \[K=x^5+y^5-x^4y-xy^4+x^2+4x+7\] For which values of $x,y$ does $K$ take its minimum value?

2005 Grigore Moisil Urziceni, 3

Tags: real analysis , limit , sequence

Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality $$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$ for all natural numbers $ n. $ [b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $ [b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $

2011 Belarus Team Selection Test, 4

Tags: system of equations , algebra

Given nonzero real numbers a,b,c with $a+b+c=a^2+b^2+c^2=a^3+b^3+c^3$. ($*$) a) Find $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)$ b) Do there exist pairwise different nonzero $a,b,c$ satisfying ($*$)? D. Bazylev

2022 China Team Selection Test, 4

Tags: geometry , incenter

Let $ABC$ be an acute triangle with $\angle ACB>2 \angle ABC$. Let $I$ be the incenter of $ABC$, $K$ is the reflection of $I$ in line $BC$. Let line $BA$ and $KC$ intersect at $D$. The line through $B$ parallel to $CI$ intersects the minor arc $BC$ on the circumcircle of $ABC$ at $E(E \neq B)$. The line through $A$ parallel to $BC$ intersects the line $BE$ at $F$. Prove that if $BF=CE$, then $FK=AD$.

2018 Polish Junior MO Finals, 1

Tags: perfect square , power , number theory

Positive odd integers $a, b$ are such that $a^bb^a$ is a perfect square. Show that $ab$ is a perfect square.

2017 Saudi Arabia BMO TST, 1

Tags: product , prime , right triangle , integer sidelengths , number theory , geometry

Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)

1995 Grosman Memorial Mathematical Olympiad, 2

Tags: winning strategy , combinatorics , game , game strategy

Two players play a game on an infinite board that consists of unit squares. Player $I$ chooses a square and marks it with $O$. Then player $II$ chooses another square and marks it with $X$. They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If no player can achieve this, the result of the game is a tie. Show that player $II$ can prevent player $I$ from winning.

1984 Balkan MO, 4

Tags: algebra

Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system: \[ ax+by=(x-y)^{2} \\ by+cz=(y-z)^{2} \\ cz+ax=(z-x)^{2}\]

1994 Mexico National Olympiad, 1

Tags: integer sequence , number theory

The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$.

1974 Dutch Mathematical Olympiad, 2

Tags: divide , divisible , number theory

$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

2003 Pan African, 3

Tags: function

Find all functions $f: R\to R$ such that: \[ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R \]