Olympiad problems

2015 Belarus Team Selection Test, 2

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

2016 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequalities

Let a,b,c be real numbers such that:$a\ge b\ge 1\ge c\ge 0$ and a+b+c=3. a)Prove that $2\le ab +bc+ca\le 3$ b)Prove that $\dfrac{24}{a^3+b^3+c^3}+\dfrac{25}{ab+bc+ca}\ge 14$. Find the equality cases

2019 Pan-African, 5

Tags: combinatorics , broken line

A square is divided into $N^2$ equal smaller non-overlapping squares, where $N \geq 3$. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). [list] [*] Show that it is possible to find a broken line composed of $4$ segments for $N = 3$. [*] Find the minimum number of segments in this broken line for arbitrary $N$. [/list]

2002 Germany Team Selection Test, 3

Tags: number theory

Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$

2000 Tuymaada Olympiad, 6

Tags: square , geometry , circumcircle , right triangle

Let $O$ be the center of the circle circumscribed around the the triangle $ABC$. The centers of the circles circumscribed around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.

1967 IMO Shortlist, 6

Tags: geometry , point set , sequence

Given a segment $AB$ of the length 1, define the set $M$ of points in the following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$

2020 JHMT, 6

Tags: geometry

Triangle $ABC$ has $\angle A = 60^o$, $\angle B = 45$, and $AC = 6$. Let $D$ be on $AB$ such that $AD = 3$. There is exactly one point $E$ on $BC$ such that $\overline{DE}$ divides $ABC$ into two cyclic polygons. Compute $DE^2$.

1996 Austrian-Polish Competition, 5

Tags: geometry , 3d geometry , sphere , polyhedron , convex

A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.

2008 Czech-Polish-Slovak Match, 1

Tags: system of equations , algebra

Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations \[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]

2012 SEEMOUS, Problem 3

Tags: matrix , linear algebra

a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$, then $$A^n=\operatorname{tr}(A)A^{n-1}.$$ b) Does the assertion from a) also hold for odd positive integers $k$?

2023 Taiwan TST Round 2, 3

Tags: geometry

Let $\Omega$ be the circumcircle of an acute triangle $ABC$. Points $D$, $E$, $F$ are the midpoints of the inferior arcs $BC$, $CA$, $AB$, respectively, on $\Omega$. Let $G$ be the antipode of $D$ in $\Omega$. Let $X$ be the intersection of lines $GE$ and $AB$, while $Y$ the intersection of lines $FG$ and $CA$. Let the circumcenters of triangles $BEX$ and $CFY$ be points $S$ and $T$, respectively. Prove that $D$, $S$, $T$ are collinear. [i]Proposed by kyou46 and Li4.[/i]

2013 Saudi Arabia IMO TST, 3

Tags: digit , divisor , number theory

For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.

2004 South africa National Olympiad, 3

Tags: floor function , ceiling function , function , algebra , number theory

Find all real numbers $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=88$. The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.

2024 Belarusian National Olympiad, 10.2

Tags: combinatorics

Some vertices of a regular $2024$-gon are marked such that for any regural polygon, all of whose vertices are vertices of the $2024$-gon, at least one of his vertices is marked. Find the minimal possible number of marked vertices [i]A. Voidelevich[/i]

2006 Harvard-MIT Mathematics Tournament, 5

Tags: geometry

Triangle $ABC$ has side lengths $AB=2\sqrt{5}$, $BC=1$, and $CA=5$. Point $D$ is on side $AC$ such that $CD=1$, and $F$ is a point such that $BF=2$ and $CF=3$. Let $E$ be the intersection of lines $AB$ and $DF$. Find the area of $CDEB$.

2024 Lusophon Mathematical Olympiad, 1

Tags: geometric sequence , algebra

Determine all geometric progressions such that the product of the three first terms is $64$ and the sum of them is $14$.

2015 Geolympiad Summer, 2.

Tags:

Let $ABC$ be a triangle. Let line $\ell$ be the line through the tangency points that are formed when the tangents from $A$ to the circle with diameter $BC$ are drawn. Let line $m$ be the line through the tangency points that are formed when the tangents from $B$ to the circle with diameter $AC$ are drawn. Show that the $\ell$, $m$, and the $C$-altitude concur.

1963 AMC 12/AHSME, 15

Tags: ratio , geometry

A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is: $\textbf{(A)}\ \sqrt{3}:1 \qquad \textbf{(B)}\ \sqrt{3}:\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{3}:2 \qquad \textbf{(D)}\ 3:\sqrt{2} \qquad \textbf{(E)}\ 3:2\sqrt{2}$

1994 Bundeswettbewerb Mathematik, 2

Tags: game , game strategy , combinatorics , algebra

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

2015 MMATHS, 1

Tags: combinatorics , algebra

Each lattice point of the plane is labeled by a positive integer. Each of these numbers is the arithmetic mean of its four neighbors (above, below, left, right). Show that all the numbers are equal.

2002 AMC 12/AHSME, 18

Tags: inequalities , algebra

If $a,b,c$ are real numbers such that $a^2+2b=7$, $b^2+4c=-7$, and $c^2+6a=-14$, find $a^2+b^2+c^2$. $\textbf{(A) }14\qquad\textbf{(B) }21\qquad\textbf{(C) }28\qquad\textbf{(D) }35\qquad\textbf{(E) }49$

2014 Mexico National Olympiad, 3

Tags: parallelogram , geometric transformation , homothety , geometry

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.

2014 China Northern MO, 5

Tags: geometry , equal angles , parallelogram

As shown in the figure, in the parallelogram $ABCD$, $I$ is the incenter of $\vartriangle BCD$, and $H$ is the orthocenter of $\vartriangle IBD$. Prove that $\angle HAB=\angle HAD$. [img]https://cdn.artofproblemsolving.com/attachments/4/3/5fa16c208ef3940443854756ae7bdb9c4272ed.png[/img]

2017 Taiwan TST Round 2, 3

Tags: number theory , function , divisibility

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2015 AMC 10, 4

Tags:

Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia? $ \textbf{(A) }\dfrac{1}{12}\qquad\textbf{(B) }\dfrac{1}{6}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad\textbf{(E) }\dfrac{1}{2} $