Found problems: 606
2015 Chile TST Ibero, 2
In the country of Muilejistan, there exists a network of roads connecting all its cities. The network has the particular property that for any two cities, there is a unique path without backtracking (i.e., a path where the traveler never returns along the same road).
The longest possible path between two cities is 600 kilometers. For instance, the path from the city of Mlar to the city of Nlar is 600 kilometers. Similarly, the path from the city of Klar to the city of Glar is also 600 kilometers.
1. If Jalim departs from Mlar towards Nlar at noon and Kalim departs from Klar towards Glar also at noon, both traveling at the same speed, prove that they meet at some point on their journey.
2. If the distance in kilometers between any two cities is an integer, prove that the distance from Glar to Mlar is even.
2024 Chile TST IMO, 3
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
2023 Germany Team Selection Test, 2
A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)
2013 Chile TST Ibero, 3
The incircle of triangle $\triangle ABC$ touches $AC$ and $BC$ at $E$ and $D$ respectively. The excircle corresponding to $A$ touches the extensions of $BC$ at $A_1$, $CA$ at $B_1$, and $AB$ at $C_1$. Let $DE \cap A_1B_1 = L$. Prove that $L$ belongs to the circumcircle of triangle $\triangle A_1B_1C_1$.
2020 USA IMO Team Selection Test, 6
Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$.
Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic.
[i]Michael Ren[/i]
2025 Turkey EGMO TST, 4
Find all positive integers $n$ such that the number
\[
\frac{3 + \sqrt{4n + 9}}{2}
\]
is the sixth smallest positive divisor of $n$.
2018 Tajikistan Team Selection Test, 3
Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression:
(b-a)(b^3+3a^3)
2018 Macedonia JBMO TST, 2
We are given a semicircle $k$ with center $O$ and diameter $AB$. Let $C$ be a point on $k$ such that $CO \bot AB$. The bisector of $\angle ABC$ intersects $k$ at point $D$. Let $E$ be a point on $AB$ such that $DE \bot AB$ and let $F$ be the midpoint of $CB$. Prove that the quadrilateral $EFCD$ is cyclic.
2021 Junior Balkan Team Selection Tests - Romania, P1
Let $n\geq 2$ be a positive integer and let $a_1,a_2,...,a_n\in[0,1]$ be real numbers. Find the maximum value of the smallest of the numbers: \[a_1-a_1a_2, \ a_2-a_2a_3,...,a_n-a_na_1.\]
2023 Israel TST, P2
Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums
\[a_{k_1}\]
\[a_{k_1}+a_{k_2}\]
\[a_{k_1}+a_{k_2}+a_{k_3}\]
\[\vdots\]
\[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\]
have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.
2022 Indonesia TST, C
Distinct pebbles are placed on a $1001 \times 1001$ board consisting of $1001^2$ unit tiles, such that every unit tile consists of at most one pebble. The [i]pebble set[/i] of a unit tile is the set of all pebbles situated in the same row or column with said unit tile. Determine the minimum amount of pebbles that must be placed on the board so that no two distinct tiles have the same [i]pebble set[/i].
[hide=Where's the Algebra Problem?]It's already posted [url=https://artofproblemsolving.com/community/c6h2742895_simple_inequality]here[/url].[/hide]
2015 Chile TST Ibero, 3
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.
2021 China Team Selection Test, 3
Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that
$$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$.
2013 Serbia Additional Team Selection Test, 3
Let $p > 3$ be a given prime number. For a set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{N}$ , define
$S_a = \{ x \in \{ 0,1, 2,...,p-1 \}$ | $(\exists_s \in S) x \equiv_p a \cdot s \}$ .
$(a)$ How many sets $S \subseteq \{ 1, 2,...,p-1 \} $ are there for which the sequence
$S_1 , S_2 , ..., S_{p-1}$ contains exactly two distinct terms?
$(b)$ Determine all numbers $k \in \mathbb{N}$ for which there is a set $ S \subseteq \{ 1, 2,...,p-1 \} $ such
that the sequence $S_1 , S_2 , ..., S_{p-1} $ contains exactly $k$ distinct terms.
[i]Proposed by Milan Basic and Milos Milosavljevic[/i]
2024 Moldova Team Selection Test, 3
Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?
2021 Romania Team Selection Test, 1
Let $k>1$ be a positive integer. A set $S{}$ is called [i]good[/i] if there exists a colouring of the positive integers with $k{}$ colours, such that no element from $S{}$ can be written as the sum of two distinct positive integers having the same colour. Find the greatest positive integer $t{}$ (in terms of $k{}$) for which the set \[S=\{a+1,a+2,\ldots,a+t\}\]is good, for any positive integer $a{}$.
2013 Dutch BxMO/EGMO TST, 4
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]
2017 Turkey Team Selection Test, 9
Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called [i]discrete coloring[/i] if there exists a circle which encloses all red points and excludes all white points. Determine the number of [i]discrete colorings[/i] for each set $S$.
2013 North Korea Team Selection Test, 3
Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.
2023 Serbia Team Selection Test, P4
Let $p$ be a prime and $P\in \mathbb{R}[x]$ be a polynomial of degree less than $p-1$ such that $\lvert P(1)\rvert=\lvert P(2)\rvert=\ldots=\lvert P(p)\rvert$. Prove that $P$ is constant.
2021 Romania Team Selection Test, 3
Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ \[s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}\]verifies the inequality \[0\leq \alpha-2ns_n\leq\frac{2}{n+1}\] for any $n\geq 2.$
2017 Turkey EGMO TST, 3
For all positive real numbers $x,y,z$ satisfying the inequality $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,$$ prove that
$$\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.$$
2020 Azerbaijan IZHO TST, 6
Define a sequence ${{a_n}}_{n\ge1}$ such that $a_1=1$ , $a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $gcd(m,a_n)\neq{1}$. Show that all positive integers occur in the sequence.
2009 Junior Balkan Team Selection Tests - Moldova, 2
Real positive numbers $a, b, c$ satisfy $abc=1$. Prove the inequality $$\frac{a^2+b^2}{a^4+b^4}+\frac{b^2+c^2}{b^4+c^4}+\frac{c^2+a^2}{c^4+a^4}\leq a+b+c.$$
2023 Serbia Team Selection Test, P3
The positive integers are partitioned into 2 sequences $a_1<a_2<\dots$ and $b_1<b_2<\dots$ such that $b_n=a_n+n$ for every positive integer $n$.
Show that $a_n+b_n=a_{b_n}$.