Olympiad problems

2008 JBMO Shortlist, 5

Tags: JBMO , geometry

Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to ${{30}^{o}}$? (The triangles may not overlap and may not exceed the margins of the square.)

2014 JBMO Shortlist, 5

Tags: geometry , JBMO

Let $ABC$ be a triangle with ${AB\ne BC}$; and let ${BD}$ be the internal bisector of $\angle ABC,\ $, $\left( D\in AC \right)$. Denote by ${M}$ the midpoint of the arc ${AC}$ which contains point ${B}$. The circumscribed circle of the triangle ${\vartriangle BDM}$ intersects the segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of ${A}$ with respect to ${K}$. If ${DJ\cap AM=\left\{O\right\}}$, prove that the points ${J, B, M, O}$ belong to the same circle.

JBMO Geometry Collection, 2003

Tags: geometry , circumcircle , angle bisector , JBMO

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

2007 JBMO Shortlist, 4

Tags: JBMO , number theory , remainder

Let $a, b$ be two co-prime positive integers. A number is called [i]good [/i] if it can be written in the form $ax + by$ for non-negative integers $x, y$. Define the function $f : Z\to Z $as $f(n) = n - n_a - n_b$, where $s_t$ represents the remainder of $s$ upon division by $t$. Show that an integer $n$ is [i]good [/i]if and only if the infinite sequence $n, f(n), f(f(n)), ...$ contains only non-negative integers.

2016 JBMO Shortlist, 1

Tags: JBMO , number theory , prime

Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.

2012 Junior Balkan MO, 2

Tags: geometry , JBMO

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

2011 JBMO Shortlist, 1

Tags: geometry , JBMO

Let $ABC$ be an isosceles triangle with $AB=AC$. On the extension of the side ${CA}$ we consider the point ${D}$ such that ${AD<AC}$. The perpendicular bisector of the segment ${BD}$ meets the internal and the external bisectors of the angle $\angle BAC$ at the points ${E}$and ${Z}$, respectively. Prove that the points ${A, E, D, Z}$ are concyclic.

2003 JBMO Shortlist, 7

Tags: geometry , circumcircle , angle bisector , JBMO

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

2020 Junior Balkаn MO, 1

Tags: algebra , system of equations , Junior Balkan , polynomial , Junior , JBMO , jbmo2020

Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2008 JBMO Shortlist, 5

Tags: JBMO , algebra , system of equations

Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system $\begin{cases} \frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\ x + y + z \le 12 \end{cases}$

2023 JBMO Shortlist, N4

Tags: JBMO , JBMO Shortlist , number theory

The triangle $ABC$ is sectioned by $AD,BE$ and $CF$ (where $D \in (BC), E \in (CA)$ and $F \in (AB)$) in seven disjoint polygons named [i]regions[/i]. In each one of the nine vertices of these regions we write a digit, such that each nonzero digit appears exactly once. We assign to each side of a region the lowest common multiple of the digits at its ends, and to each region the greatest common divisor of the numbers assigned to its sides. Find the largest possible value of the product of the numbers assigned to the regions.

2013 JBMO Shortlist, 1

Tags: geometry , JBMO

Let ${AB}$ be a diameter of a circle ${\omega}$ and center ${O}$ , ${OC}$ a radius of ${\omega}$ perpendicular to $AB$,${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points ${M,O,P,N}$ are cocyclic. (Albania)

2008 JBMO Shortlist, 4

Tags: JBMO , algebra , system of equations

Find all triples $(x,y,z)$ of real numbers that satisfy the system $\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}$

2008 JBMO Shortlist, 8

Tags: JBMO , number theory

Let $a, b, c, d, e, f$ are nonzero digits such that the natural numbers $\overline{abc}, \overline{def}$ and $\overline{abcdef }$ are squares. a) Prove that $\overline{abcdef}$ can be represented in two different ways as a sum of three squares of natural numbers. b) Give an example of such a number.

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , algebra , 2019 , JBMO , romania

Find the maximum value of: $E(a,b)=\frac{a+b}{(4a^2+3)(4b^2+3)}$ For $a,b$ real numbers.

2009 JBMO Shortlist, 4

Tags: JBMO , number theory , Sum of powers

Determine all prime numbers $p_1, p_2,..., p_{12}, p_{13}, p_1 \le p_2 \le ... \le p_{12} \le p_{13}$, such that $p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2$ and one of them is equal to $2p_1 + p_9$.

2003 JBMO Shortlist, 3

Tags: JBMO , geometry

Let $G$ be the centroid of triangle $ABC$, and $A'$ the symmetric of $A$ wrt $C$. Show that $G, B, C, A'$ are concyclic if and only if $GA \perp GC$.

2007 JBMO Shortlist, 1

Tags: least common multiple , greatest common divisor , JBMO , number theory

Find all the pairs positive integers $(x, y)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}$ , where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.

2010 JBMO Shortlist, 3

Tags: geometry , JBMO

Consider a triangle ${ABC}$ and let ${M}$ be the midpoint of the side ${BC.}$ Suppose ${\angle MAC=\angle ABC}$ and ${\angle BAM=105^{\circ}.}$ Find the measure of ${\angle ABC}$.

2011 JBMO Shortlist, 9

Tags: JBMO , combinatorics , combinatorial geometry

Decide if it is possible to consider $2011$ points in a plane such that the distance between every two of these points is different from $1$ and each unit circle centered at one of these points leaves exactly $1005$ points outside the circle.

2016 JBMO Shortlist, 7

Tags: geometry , JBMO

Let ${AB}$ be a chord of a circle ${(c)}$ centered at ${O}$, and let ${K}$ be a point on the segment ${AB}$ such that ${AK<BK}$. Two circles through ${K}$, internally tangent to ${(c)}$ at ${A}$ and ${B}$, respectively, meet again at ${L}$. Let ${P}$ be one of the points of intersection of the line ${KL}$ and the circle ${(c)}$, and let the lines ${AB}$ and ${LO}$ meet at ${M}$. Prove that the line ${MP}$ is tangent to the circle ${(c)}$. Theoklitos Paragyiou (Cyprus)

2008 JBMO Shortlist, 11

Tags: JBMO , number theory

Determine the greatest number with $n$ digits in the decimal representation which is divisible by $429$ and has the sum of all digits less than or equal to $11$.

2003 JBMO Shortlist, 6

Tags: inequalities , JBMO , JBMO Shortlist

Parallels to the sides of a triangle passing through an interior point divide the inside of a triangle into $6$ parts with the marked areas as in the figure. Show that $\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}$ [img]https://cdn.artofproblemsolving.com/attachments/a/a/b0a85df58f2994b0975b654df0c342d8dc4d34.png[/img]

2018 JBMO Shortlist, NT1

Tags: JBMO , number theory , algebra

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

2011 JBMO Shortlist, 5

Tags: geometry , JBMO

Inside the square ${ABCD}$, the equilateral triangle $\vartriangle ABE$ is constructed. Let ${M}$ be an interior point of the triangle $\vartriangle ABE$ such that $MB=\sqrt{2}$, $MC=\sqrt{6}$, $MD=\sqrt{5}$ and ${ME=\sqrt{3}}$. Find the area of the square ${ABCD}$.