Found problems: 177
2018 Junior Balkan MO, 1
Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.
2015 JBMO Shortlist, 1
Around the triangle $ABC$ the circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to this tangent intersects the lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$, respectively. Prove that the points $A,B,D,E$ belong to the same circle.
(Montenegro)
2007 JBMO Shortlist, 3
Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.
2007 JBMO Shortlist, 1
We call a tiling of an $m \times n$ rectangle with corners (see figure below) "regular" if there is no sub-rectangle which is tiled with corners. Prove that if for some $m$ and $n$ there exists a "regular" tiling of the $m \times n$ rectangular then there exists a "regular" tiling also for the $2m \times 2n $ rectangle.
2019 Junior Balkan MO, 2
Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that
$a^4 - 2019a = b^4 - 2019b = c$.
Prove that $- \sqrt{c} < ab < 0$.
2012 JBMO ShortLists, 1
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
\[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\]
When does equality hold?
2003 JBMO Shortlist, 4
Three equal circles have a common point $M$ and intersect in pairs at points $A, B, C$. Prove that that $M$ is the orthocenter of triangle $ABC$.
2008 JBMO Shortlist, 2
For a fixed triangle $ABC$ we choose a point $M$ on the ray $CA$ (after $A$), a point $N$ on the ray $AB$ (after $B$) and a point $P$ on the ray $BC$ (after $C$) in a way such that $AM -BC = BN- AC = CP – AB$. Prove that the angles of triangle $MNP$ do not depend on the choice of $M, N, P$ .
2003 Junior Balkan MO, 4
Let $x, y, z > -1$. Prove that \[ \frac{1+x^2}{1+y+z^2} + \frac{1+y^2}{1+z+x^2} + \frac{1+z^2}{1+x+y^2} \geq 2. \]
[i]Laurentiu Panaitopol[/i]
2011 JBMO Shortlist, 1
Inside of a square whose side length is $1$ there are a few circles such that the sum of their circumferences is equal to $10$. Show that there exists a line that meets at least four of these circles.
2011 JBMO Shortlist, 3
We can change a natural number $n$ in three ways:
a) If the number $n$ has at least two digits, we erase the last digit and we subtract that digit from the remaining number (for example, from $123$ we get $12 - 3 = 9$);
b) If the last digit is different from $0$, we can change the order of the digits in the opposite one (for example, from $123$ we get $321$);
c) We can multiply the number $n$ by a number from the set $ \{1, 2, 3,..., 2010\}$.
Can we get the number $21062011$ from the number $1012011$?
2015 Junior Balkan Team Selection Tests - Romania, 3
Prove that if $a,b,c>0$ and $a+b+c=1,$ then $$\frac{bc+a+1}{a^2+1}+\frac{ca+b+1}{b^2+1}+\frac{ab+c+1}{c^2+1}\leq \frac{39}{10}$$
2008 JBMO Shortlist, 5
Is it possible to arrange the numbers $1^1, 2^2,..., 2008^{2008}$ one after the other, in such a way that the obtained number is a perfect square? (Explain your answer.)
2009 JBMO Shortlist, 2
A group of $n > 1$ pirates of different age owned total of $2009$ coins. Initially each pirate (except the youngest one) had one coin more than the next younger.
a) Find all possible values of $n$.
b) Every day a pirate was chosen. The chosen pirate gave a coin to each of the other pirates. If $n = 7$, find the largest possible number of coins a pirate can have after several days.
2008 JBMO Shortlist, 7
Let $a, b$ and $c$ be positive real numbers such that $abc = 1$. Prove the inequality
$\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c)$.
2011 JBMO Shortlist, 3
Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$, ${AH}$ is an altitude of$\vartriangle ABC$ $\left( H\in BC \right)$ and ${M}$is the midpoint of the side ${AB}$. It is known that the midpoints of the segments ${BL}$ and ${MH}$ coincides. Determine the internal angles of triangle $\vartriangle ABC$.
2013 JBMO Shortlist, 3
Show that
\[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\]
for all positive real numbers $a$ and $b$ such that $ab\geq 1$.
2005 JBMO Shortlist, 7
Let $ABCD$ be a parallelogram. $P \in (CD), Q \in (AB)$, $M= AP \cap DQ$, $N=BP \cap CQ$, $ K=MN \cap AD$, $L= MN \cap BC$. Prove that $BL=DK$.
2009 JBMO Shortlist, 3
a) In how many ways can we read the word SARAJEVO from the table below, if it is allowed to jump from cell to an adjacent cell (by vertex or a side) cell?
b) After the letter in one cell was deleted, only $525$ ways to read the word SARAJEVO remained. Find all possible positions of that cell.
2004 JBMO Shortlist, 4
Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC], E\in [BC]$. Inside the triangle we construct the semicircles $C_1, C_2, C_3, C_4$ of diameters $[AC], [BC], [CD], [CE]$ and let $\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4$. Show that points $K, L, M, N$ are concyclic.
2008 JBMO Shortlist, 6
Let $ABC$ be a triangle with $\angle A<{{90}^{o}} $. Outside of a triangle we consider isosceles triangles $ABE$ and $ACZ$ with bases $AB$ and $AC$, respectively. If the midpoint $D$ of the side $BC$ is such that $DE \perp DZ$ and $EZ = 2 \cdot ED$, prove that $\angle AEB = 2 \cdot \angle AZC$ .
2016 JBMO Shortlist, 1
Let ${ABC}$ be an acute angled triangle, let ${O}$ be its circumcentre, and let ${D,E,F}$ be points on the sides ${BC,CA,AB}$, respectively. The circle ${(c_1)}$ of radius ${FA}$, centered at ${F}$, crosses the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centered at $D$, crosses the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$. Finally, the circle ${(c_3)}$ of radius $EC$, centered at $E$, crosses the segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$. Prove that the quadrilaterals $BKF{A}',CLD{B}'$ and $AME{C}'$ are all cyclic, and their circumcircles share a common point.
Evangelos Psychas (Greece)
2011 JBMO Shortlist, 5
A set $S$ of natural numbers is called [i]good[/i], if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a [i]good [/i]set which is a subset of the set $A = \{1,2, 3, ...,63\}$.
2011 JBMO Shortlist, 5
Find the least positive integer such that the sum of its digits is $2011$ and the product of its digits is a power of $6$.
2003 JBMO Shortlist, 1
Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?