Olympiad problems

2011 All-Russian Olympiad, 2

Tags: geometry

Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extensions, points $P$ and $Q$ are chosen, such that angle $PAQ$ is right. Let $AF$ be a height of triangle $APQ$. Prove that angle $BFC$ is a right angle.

2016 Czech-Polish-Slovak Junior Match, 2

Tags: divide , digit , number theory

Find the largest integer $d$ divides all three numbers $abc, bca$ and $cab$ with $a, b$ and $c$ being some nonzero and mutually different digits. Czech Republic

ICMC 8, 1

Tags:

Joe the Jaguar is on an infinite grid of unit squares, starting at the centre of one of them. At the $k$-th minute, Joe must jump a distance of $k$ units in any direction. For which $n$ is it possible for Joe to be inside or on the edge of the starting square after $n$ minutes?

2004 Bundeswettbewerb Mathematik, 4

Tags: inequalities , calculus , 3d geometry , sphere , geometry

A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.

2007 Turkey Team Selection Test, 1

Tags: analytic geometry , circumcircle , geometry , trigonometry

[color=indigo]Let $ABC$ is an acute angled triangle and let $A_{1},\, B_{1},\, C_{1}$ are points respectively on $BC,\,CA,\,AB$ such that $\triangle ABC$ is similar to $\triangle A_{1}B_{1}C_{1}.$ Prove that orthocenter of $A_{1}B_{1}C_{1}$ coincides with circumcenter of $ABC$.[/color]

2024 Kyiv City MO Round 1, Problem 1

Tags: geometry , rectangle

Square $ABCD$ is cut by a line segment $EF$ into two rectangles $AEFD$ and $BCFE$. The lengths of the sides of each of these rectangles are positive integers. It is known that the area of the rectangle $AEFD$ is $30$ and it is larger than the area of the rectangle $BCFE$. Find the area of square $ABCD$. [i]Proposed by Bogdan Rublov[/i]

2009 Princeton University Math Competition, 7

Tags: inequalities

Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.

2018 Israel Olympic Revenge, 4

Tags: function , algebra , functional equation

Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that $$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$ for all $f,g:\mathbb R\to\mathbb R$. Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that $$F(f)=\sigma\circ f\circ\sigma$$ for all $f:\mathbb R\to\mathbb R$.

1993 Turkey Team Selection Test, 3

Tags: inequalities

Let ($b_n$) be a sequence such that $b_n \geq 0 $ and $b_{n+1}^2 \geq \frac{b_1^2}{1^3}+\cdots+\frac{b_n^2}{n^3}$ for all $n \geq 1$. Prove that there exists a natural number $K$ such that \[\sum_{n=1}^{K} \frac{b_{n+1}}{b_1+b_2+ \cdots + b_n} \geq \frac{1993}{1000}\]

1994 Romania TST for IMO, 4:

Tags: algebra

Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that $(1)$ $f(n) \leq n^8$ for any $n \geq 2$, $(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that $$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$

2015 Caucasus Mathematical Olympiad, 3

Tags: combinatorial geometry , combinatorics , tiling , tiles

The workers laid a floor of size $n \times n$ with tiles of two types: $2 \times 2$ and $3 \times 1$. It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen? (You can’t cut tiles and also put them on top of each other.)

2018 Pan African, 2

Tags: ratio , combinatorics

A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $\frac{7}{9}$. How many players took part at the tournament?

2021 South East Mathematical Olympiad, 1

Tags: algebra , sequence

A sequence $\{a_n\}$ is defined recursively by $a_1=\frac{1}{2}, $ and for $n\ge 2,$ $0<a_n\leq a_{n-1}$ and \[a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0.\] $(1)$ Determine the general formula of the sequence $\{a_n\};$ $(2)$ Let $S_n=a_1+\cdots+a_n.$ Prove that for $n\ge 1,$ $\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).$

2021 Dutch IMO TST, 2

Tags: geometry , right angle , centroid , right triangle

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

2012 CHMMC Spring, 3

Tags: combinatorics

In a $ 4 \times 4 $ grid of sixteen unit squares, exactly $8$ are shaded so that each shaded square shares an edge with exactly one other shaded square. How many ways can this be done?

2022 HMNT, 6

Tags: geometry

A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected without either player winning, the game ends in a draw. Given that both players play optimally, find all positive areas of the convex polygon formed by Alice's points at the end of the game.

2014 Contests, 3

Tags: geometry , perpendicular , diagonal

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

2019 Bulgaria National Olympiad, 5

Tags: combinatorial geometry , combinatorics , geometry

Let $P$ be a $2019-$gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)

2014 India PRMO, 7

Tags: algebra , exponential

If $x^{x^4}=4 $ what is the value of $x^{x^2}+x^{x^8} $ ?

1994 Baltic Way, 8

Tags: number theory

Show that for any integer $a\ge 5$ there exist integers $b$ and $c$, $c\ge b\ge a$, such that $a,b,c$ are the lengths of the sides of a right-angled triangle.

1989 Federal Competition For Advanced Students, 2

Tags: inequalities

If $ a$ and $ b$ are nonnegative real numbers with $ a^2\plus{}b^2\equal{}4$, show that: $ \frac{ab}{a\plus{}b\plus{}2} \le \sqrt{2}\minus{}1$ and determine when equality occurs.

2018 Saudi Arabia JBMO TST, 1

Tags: number theory

$p, q, r$ are distinct prime numbers which satisfy $$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$ for natural number $A$. Find all values of $A$.

2019 Switzerland Team Selection Test, 12

Tags: number theory

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

2014 Contests, 1

Tags: polynomial , induction , algebra

Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

1992 IMO Longlists, 25

Tags: combinatorics , partition , coloring , ramsey theory

[b][i](a) [/i][/b] Show that the set $\mathbb N$ of all positive integers can be partitioned into three disjoint subsets $A, B$, and $C$ satisfying the following conditions: \[A^2 = A, B^2 = C, C^2 = B,\] \[AB = B, AC = C, BC = A,\] where $HK$ stands for $\{hk | h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb N$, and $H^2$ denotes $HH.$ [b][i](b)[/i][/b] Show that for every such partition of $\mathbb N$, $\min\{n \in N | n \in A \text{ and } n + 1 \in A\}$ is less than or equal to $77.$