Found problems: 109
2019 JBMO Shortlist, C4
We have a group of $n$ kids. For each pair of kids, at least one has sent a message to
the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly
$25 \% $ have sent a message to $A$. How many possible two-digit values of $n$ are there?
[i]Proposed by Bulgaria[/i]
2022 JBMO Shortlist, G1
Let $ABCDE$ be a cyclic pentagon such that $BC = DE$ and $AB$ is parallel to $DE$. Let $X, Y,$ and $Z$ be the midpoints of $BD, CE,$ and $AE$ respectively. Show that $AE$ is tangent to the circumcircle of the triangle $XYZ$.
Proposed by [i]Nikola Velov, Macedonia[/i]
2021 JBMO Shortlist, N5
Find all pairs of integers $(x, y)$ such that $x^2 + 5y^2 = 2021y$.
2023 JBMO Shortlist, G2
Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.
2013 Junior Balkan MO, 4
Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses $n$ real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a. $n=5$
b. $n=6$
c. $n=8$
Justify your answer(s).
[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]
2022 JBMO Shortlist, A1
Find all pairs of positive integers $(a, b)$ such that $$11ab \le a^3 - b^3 \le 12ab.$$
2014 Junior Balkan MO, 3
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
2013 JBMO Shortlist, 3
Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses $n$ real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a. $n=5$
b. $n=6$
c. $n=8$
Justify your answer(s).
[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]
2023 JBMO Shortlist, C1
Given is a square board with dimensions $2023 \times 2023$, in which each unit cell is colored blue or red. There are exactly $1012$ rows in which the majority of cells are blue, and exactly $1012$ columns in which the majority of cells are red.
What is the maximal possible side length of the largest monochromatic square?
2024 Junior Balkan Team Selection Tests - Romania, P1
For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that
$$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$
2023 JBMO Shortlist, G5
Let $D,E,F$ be the points of tangency of the incircle of a given triangle $ABC$ with sides $BC, CA, AB,$ respectively. Denote by $I$ the incenter of $ABC$, by $M$ the midpoint of $BC$ and by $G$ the foot of the perpendicular from $M$ to line $EF$. Prove that the line $ID$ is tangent to the circumcircle of the triangle $MGI$.
2014 JBMO Shortlist, 3
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
2024 Junior Balkan Team Selection Tests - Moldova, 10
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that
$$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$
2023 JBMO Shortlist, G7
Let $D$ and $E$ be arbitrary points on the sides $BC$ and $AC$ of triangle $ABC$, respectively. The circumcircle of $\triangle ADC$ meets for the second time the circumcircle of $\triangle BCE$ at point $F$. Line $FE$ meets line $AD$ at point $G$, while line $FD$ meets line $BE$ at point $H$. Prove that lines $CF, AH$ and $BG$ pass through the same point.
2013 JBMO Shortlist, 2
Solve in integers $20^x+13^y=2013^z$.
2023 JBMO Shortlist, A3
Prove that for all non-negative real numbers $x,y,z$, not all equal to $0$, the following inequality holds
$\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.$
Determine all the triples $(x,y,z)$ for which the equality holds.
[i]Milan Mitreski, Serbia[/i]
2022 JBMO Shortlist, C5
Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$.
Proposed by [i]Viktor Simjanoski, Macedonia[/i]
2022 Greece JBMO TST, 1
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to
factorials of some positive integers.
Proposed by [i]Nikola Velov, Macedonia[/i]
2022 Junior Balkan Mathematical Olympiad, 2
Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.
2020 JBMO Shortlist, 1
Alice and Bob play the following game: starting with the number $2$ written on a blackboard, each player in turn changes the current number $n$ to a number $n + p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than $\underbrace{22...2}_{2020}$. Assuming perfect play, who will win the game.
JBMO Geometry Collection, 2021
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$.
Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.
2021 Junior Balkan Team Selection Tests - Moldova, 4
Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that
$73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.
2023 JBMO Shortlist, C4
Anna and Bob are playing the following game: The number $2$ is initially written on the blackboard. With Anna playing first, they alternately double the number currently written on the blackboard or square it.
The person who first writes on the blackboard a number greater than $2023^{10}$ is the winner. Determine which player has a winning strategy.
2022 JBMO Shortlist, N3
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$
2022 JBMO Shortlist, G3
Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.