Found problems: 19
2016 Junior Balkan Team Selection Test, 2
Find minimal number of divisors that can number $|2016^m-36^n|$ have,where $m$ and $n$ are natural numbers.
2017 Macedonia JBMO TST, Source
[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url]
[url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.
[url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that
$$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$
Show that $TB_2AC_2$ is a rectangle.
[url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that
$$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$
When does equality happen?
[url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$
[url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.
2022 Junior Balkan Team Selection Tests - Romania, P4
Let $a,b,c>0$ such that $a+b+c=3$. Prove that :$$\frac{ab}{ab+a+b}+\frac{bc}{bc+b+c}+\frac{ca}{ca+c+a}+\frac{1}{9}\left(\frac{(a-b)^2}{ab+a+b}+\frac{(b-c)^2}{bc+b+c}+\frac{(c-a)^2}{ca+c+a}\right)\leq1.$$
2022 Bosnia and Herzegovina Junior BMO TST, 3
Let $ABC$ be an acute triangle. Tangents on the circumscribed circle of triangle $ABC$ at points $B$ and $C$ intersect at point $T$. Let $D$ and $E$ be a foot of the altitudes from $T$ onto $AB$ and $AC$ and let $M$ be the midpoint of $BC$. Prove:
A) Prove that $M$ is the orthocenter of the triangle $ADE$.
B) Prove that $TM$ cuts $DE$ in half.
2025 Junior Macedonian Mathematical Olympiad, 5
Let $M$ be the midpoint of side $BC$ in $\triangle ABC$, and $P \neq B$ is such that the quadrilateral $ABMP$ is cyclic and the circumcircle of $\triangle BPC$ is tangent to the line $AB$. If $E$ is the second common point of the line $BP$ and the circumcircle of $\triangle ABC$, determine the ratio $BE: BP$.
2022 Bosnia and Herzegovina Junior BMO TST, 1
Let $a,b,c$ be real numbers such that $$a^2-bc=b^2-ca=c^2-ab=2$$. Find the value of $$ab+bc+ca$$ and find at least one triplet $(a,b,c)$ that satisfy those conditions.
2025 Junior Macedonian Mathematical Olympiad, 4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]
When does the equality hold?
2024 Junior Macedonian Mathematical Olympiad, 4
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.
[i]Proposed by Ilija Jovčeski[/i]
2018 Serbia JBMO TST, 4
Two players are playing the following game.
They are alternatively putting blue and red coins on the board $2018$ by $2018$. If first player creates $n$ blue coins in a row or column, he wins. Second player wins if he can prevent it. Who will win if:
$a)n=4$;
$b)n=5$?
Note: first player puts only blue coins, and second only red.
2023 Bulgaria JBMO TST, 2
Determine the smallest positive integer $n\geq 2$ for which there exists a positive integer $m$ such that $mn$ divides $m^{2023} + n^{2023} + n$.
2025 Junior Macedonian Mathematical Olympiad, 2
Let $B_1$ be the foot of the altitude from the vertex $B$ in the acute-angled $\triangle ABC$. Let $D$ be the midpoint of side $AB$, and $O$ be the circumcentre of $\triangle ABC$. Line $B_1D$ meets line $CO$ at $E$. Prove that the points $B, C, B_1$, and $E$ lie on a circle.
2022 Azerbaijan JBMO TST, C4
$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)
2024 Junior Macedonian Mathematical Olympiad, 2
It is known that in a group of $2024$ students each student has at least $1011$ acquaintances among the remaining members of the group. What is more, there exists a student that has at least $1012$ acquaintances in the group. Prove that for every pair of students $X, Y$, there exist students $X_0 = X, X_1, ..., X_{n - 1}, X_n = Y$ in the group such that for every index $i = 0, ..., n - 1$, the students $X_i$ and $X_{i + 1}$ are acquaintances.
[i]Proposed by Mirko Petruševski[/i]
2017 Greece JBMO TST, Source
[url=https://artofproblemsolving.com/community/c675547][b]Greece JBMO TST 2017[/b][/url]
[url=http://artofproblemsolving.com/community/c6h1663730p10567608][b]Problem 1[/b][/url]. Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that
$$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$
Also, find the values of $a,b,c$ for which the equality happens.
[url=http://artofproblemsolving.com/community/c6h1663731p10567619][b]Problem 2[/b][/url]. Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$. The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$. Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$.
[url=http://artofproblemsolving.com/community/c6h1663732p10567627][b]Problem 3[/b][/url]. Prove that for every positive integer $n$, the number $A_n = 7^{2n} -48n - 1$ is a multiple of $9$.
[url=http://artofproblemsolving.com/community/c6h1663734p10567640][b]Problem 4[/b][/url]. Let $ABC$ be an equilateral triangle of side length $a$, and consider $D$, $E$ and $F$ the midpoints of the sides $(AB), (BC)$, and $(CA)$, respectively. Let $H$ be the the symmetrical of $D$ with respect to the line $BC$. Color the points $A, B, C, D, E, F, H$ with one of the two colors, red and blue.
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[*] How many equilateral triangles with all the vertices in the set $\{A, B, C, D, E, F, H\}$ are there?
[*] Prove that if points $B$ and $E$ are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set $\{A, B, C, D, E, F, H\}$ and having the same color.
[*] Does the conclusion of the second part remain valid if $B$ is blue and $E$ is red?
[/list]
2022 Junior Balkan Team Selection Tests - Romania, P2
Find the largest positive integer $n$ such that the following is true:
There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$.
2022 Bosnia and Herzegovina Junior BMO TST, 2
Let $a,b,c$ be positive integers greater than $1$ such that $$p=ab+bc+ac$$ is prime.
A) Prove that $a^2, b^2, c^2$ all have different reminder $mod\ p$.
B) Prove that $a^3, b^3, c^3$ all have different reminder $mod\ p$.
2025 Junior Macedonian Mathematical Olympiad, 3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
2022 Bosnia and Herzegovina Junior BMO TST, 4
Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn't know at least $20$ other persons and at least $2$ of those $20$ don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any $social$ nor $shy$ persons.
2025 Junior Macedonian Mathematical Olympiad, 1
Batman, Robin, and The Joker are in three of the vertex cells in a square $2025 \times 2025$ board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker.
If in each move all three can see where the others moved, who has a winning strategy, The Joker, or Batman and Robin? Explain the answer.
[b]Comment.[/b] Batman and Robin decide their common strategy at the beginning.
[img]https://i.imgur.com/PeLBQNt.png[/img]