Olympiad problems

1979 Spain Mathematical Olympiad, 7

Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.

2015 Belarus Team Selection Test, 2

Tags: algebra , function , calculus

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]