Found problems: 606
2023 Israel TST, P2
For each positive integer $n$, define $A(n)$ to be the sum of its divisors, and $B(n)$ to be the sum of products of pairs of its divisors. For example,
\[A(10)=1+2+5+10=18\]
\[B(10)=1\cdot 2+1\cdot 5+1\cdot 10+2\cdot 5+2\cdot 10+5\cdot 10=97\]
Find all positive integers $n$ for which $A(n)$ divides $B(n)$.
2023 Azerbaijan JBMO TST, 3
Let $ABC$ be a triangle and let $\Omega$ denote the circumcircle of $ABC$. The foot of altitude from $A$ to $BC$ is $D$. The foot of altitudes from $D$ to $AB$ and $AC$ are $K;L$ , respectively. Let $KL$ intersect $\Omega$ at $X;Y$, and let $AD$ intersect $\Omega$ at $Z$. Prove that $D$ is the incenter of triangle $XYZ$
2023 Iran Team Selection Test, 2
Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $ a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$, Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$.
[i]Proposed by Navid Safaei [/i]
2023 Israel TST, P2
Let $ABC$ be an isosceles triangle, $AB=AC$ inscribed in a circle $\omega$. The $B$-symmedian intersects $\omega$ again at $D$. The circle through $C,D$ and tangent to $BC$ and the circle through $A,D$ and tangent to $CD$ intersect at points $D,X$. The incenter of $ABC$ is denoted $I$. Prove that $B,C,I,X$ are concyclic.
2016 Bosnia and Herzegovina Junior BMO TST, 3
Let $O$ be a center of circle which passes through vertices of quadrilateral $ABCD$, which has perpendicular diagonals. Prove that sum of distances of point $O$ to sides of quadrilateral $ABCD$ is equal to half of perimeter of $ABCD$.
1998 Brazil Team Selection Test, Problem 4
(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite.
(b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.
2019 Romania Team Selection Test, 2
Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.
2024 Chile TST Ibero., 1
Determine all integers \( x \) for which the expression \( x^2 + 10x + 160 \) is a perfect square.
2025 Turkey EGMO TST, 1
A chessboard with some unit squares marked is called a $\textit{good board}$ if for any pair of rows \((s, t)\), a rook placed on a marked square in row \(s\) can reach a marked square in row \(t\) in several moves by only moving to marked squares above, below, or to the right of its current position.
Consider a chessboard with 220 rows and 12 columns, where exactly 9 unit squares in each row are marked. Regardless of how the marked squares are chosen, if it is possible to delete \(k\) columns and rearrange the remaining columns to form a $\textit{good board}$ determine the maximum possible value of \(k\).
2023 Thailand October Camp, 6
In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
[list]
[*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
[*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.
[/list]
We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.
2016 Dutch IMO TST, 4
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.
2017 Saudi Arabia JBMO TST, 2
Find all prime numbers $p$ such that $\frac{3^{p-1} - 1}{p}$ is a perfect square.
2025 Turkey EGMO TST, 6
In a chess tournament with 200 participants, 700 matches are arranged such that among any 100 participants, the number of matches played between them is at least \( N \). Determine the maximum possible value of \( N \).
2023 Indonesia TST, 1
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
2017 Turkey Team Selection Test, 5
For all positive real numbers $a,b,c$ with $a+b+c=3$, show that
$$a^3b+b^3c+c^3a+9\geq 4(ab+bc+ca).$$
2008 China Team Selection Test, 3
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$
2025 Junior Balkan Team Selection Tests - Romania, P4
Let $ABCDEF$ be a convex hexagon, such that the triangles $ABC$ and $DEF$ are equilateral and the diagonals $AD, BE$ and $CF$ are concurrent. Prove that $AC\parallel DF$ or $BE=AD+CF.$
2018 Azerbaijan IZhO TST, 2
Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that:
a/(1-x)+b/(1-y)=1
Prove that:
∛ay+∛bx≤1.
2024 Israel TST, P3
Let $ABCD$ be a parallelogram. Let $\omega_1$ be the circle passing through $D$ tangent to $AB$ at $A$. Let $\omega_2$ be the circle passing through $A$ tangent to $CD$ at $D$. The tangents from $B$ to $\omega_1$ touch it at $A$ and $P$. The tangents from $C$ to $\omega_2$ touch it at $D$ and $Q$. Lines $AP$ and $DQ$ intersect at $X$. The perpendicular bisector of $BC$ intersects $AD$ at $R$.
Show that the circumcircles of triangles $\triangle PQX$, $\triangle BCR$ are concentric.
2024 Germany Team Selection Test, 3
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
2012 JBMO TST - Macedonia, 5
$ n\geq 4 $ points are given in a plane such that any 3 of them are not collinear. Prove that a triangle exist such that all the points are in its interior and there is exactly one point laying on each side.
2021 Azerbaijan IZhO TST, 2
Find the number of ways to color $n \times m$ board with white and black
colors such that any $2 \times 2$ square contains the same number of black and white cells.
2021 Azerbaijan IZhO TST, 4
Let $ABC$ be a triangle with incircle touching $BC, CA, AB$ at $D, E,
F,$ respectively. Let $O$ and $M$ be its circumcenter and midpoint of $BC.$ Suppose that circumcircles of $AEF$ and $ABC$ intersect at $X$ for the second time. Assume $Y \neq X$ is on the circumcircle of $ABC$ such that $OMXY$ is cyclic. Prove that circumcenter of $DXY$ lies on $BC.$
[i]Proposed by tenplusten.[/i]
2024 Indonesia TST, 2
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
2022 Bolivia Cono Sur TST, P5
Find the sum of all even numbers greater than 100000, that u can make only with the digits 0,2,4,6,8,9 without any digit repeating in any number.