Found problems: 606
2021 Federal Competition For Advanced Students, P2, 1
Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$.
When does equality holds?
(Karl Czakler)
2019 Moldova Team Selection Test, 2
Prove that $E_n=\frac{\arccos {\frac{n-1}{n}} } {\text{arccot} {\sqrt{2n-1} }}$ is a natural number for any natural number $n$.
(A natural number is a positive integer)
2024 Azerbaijan JBMO TST, 4
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 Indonesia TST, 1
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 $3$.)
2023 Azerbaijan IZhO TST, 4
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove
that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive
integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.
2020 Junior Balkan Team Selection Tests - Moldova, 12
Find all numbers $n \in \mathbb{N}^*$ for which there exists a finite set of natural numbers $A=(a_1, a_2,...a_n)$ so that for any $k$ $(1\leq k \leq n)$ the number $a_k$ is the number of all multiples of $k$ in set $A$.
2023 Balkan MO Shortlist, G6
Let $ABC$ be an acute triangle ($AB < BC < AC$) with circumcircle $\Gamma$. Assume there exists $X \in AC$ satisfying $AB=BX$ and $AX=BC$. Points $D, E \in \Gamma$ are taken such that $\angle ADB<90^{\circ}$, $DA=DB$ and $BC=CE$. Let $P$ be the intersection point of $AE$ with the tangent line to $\Gamma$ at $B$, and let $Q$ be the intersection point of $AB$ with tangent line to $\Gamma$ at $C$. Show that the projection of $D$ onto $PQ$ lies on the circumcircle of $\triangle PAB$.
2017 Junior Balkan Team Selection Tests - Romania, 5
Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that ${\angle ADB= \angle AEC=90^o}$ and ${\angle BAD= \angle CAE}$. Let ${{A}_{1}}\in BC,{{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and ${K,L}$ be the midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles $AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear.
(Bulgaria)
1986 China Team Selection Test, 3
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$
2020 USA IMO Team Selection Test, 5
Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc.
[i]Carl Schildkraut[/i]
2004 Serbia Team Selection Test, 2
Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that the most two of numbers
$$2a-\frac{1}{b},\ 2b-\frac{1}{c},\ 2c-\frac{1}{a}$$
are greater than $1$.
2021 Romania Team Selection Test, 2
Let $N\geq 4$ be a fixed positive integer. Two players, $A$ and $B$ are forming an ordered set $\{x_1,x_2,...\},$ adding elements alternatively. $A$ chooses $x_1$ to be $1$ or $-1,$ then $B$ chooses $x_2$ to be $2$ or $-2,$ then $A$ chooses $x_3$ to be $3$ or $-3,$ and so on. (at the $k^{th}$ step, the chosen number must always be $k$ or $-k$)
The winner is the first player to make the sequence sum up to a multiple of $N.$ Depending on $N,$ find out, with proof, which player has a winning strategy.
2023 Bulgaria JBMO TST, 2
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that
$$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$
Proposed by [i]Petar Filipovski, Macedonia[/i]
2024 Junior Macedonian Mathematical Olympiad, 3
The angle bisector of $\angle BAC$ intersects the circumcircle of the acute-angled $\triangle ABC$ at point $D$. Let the perpendicular bisectors of $CD$ and $AD$ intersect sides $BC$ and $AB$ at points $E$ and $F$, respectively. If $O$ is the circumcenter of $\triangle ABC$, prove that the points $F, D, E$, and $O$ are concyclic.
[i]Proposed by Petar Filipovski[/i]
2024 Argentina Cono Sur TST, 6
Find all pairs of positive integers $(n, k)$ that satisfy the equation
$$n!+n=n^k$$
2019 Slovenia Team Selection Test, 5
Let $ABC$ be a triangle and $D, E$ and $F$ the foots of heights from $A, B$ and $C$ respectively. Let $D_1$ be such a point on $EF$, that $DF = D_1 E$ where $E$ is between $D_1$ and $F$. Similarly, let $D_2$ be such a point on $EF$, that $DE = D_2 F$ where $F$ is between $E$ and $D_2$. Let the bisector of $DD_1$ intersect $AB$ at $P$ and let the bisector of $DD_2$ intersect $AC$ at $Q$.
Prove that, $PQ$ bisects $BC$.
2018 USA Team Selection Test, 1
Let $n$ be a positive integer and let $S \subseteq \{0, 1\}^n$ be a set of binary strings of length $n$. Given an odd number $x_1, \dots, x_{2k + 1} \in S$ of binary strings (not necessarily distinct), their [i]majority[/i] is defined as the binary string $y \in \{0, 1\}^n$ for which the $i^{\text{th}}$ bit of $y$ is the most common bit among the $i^{\text{th}}$ bits of $x_1, \dots,x_{2k + 1}$. (For example, if $n = 4$ the majority of 0000, 0000, 1101, 1100, 0101 is 0100.)
Suppose that for some positive integer $k$, $S$ has the property $P_k$ that the majority of any $2k + 1$ binary strings in $S$ (possibly with repetition) is also in $S$. Prove that $S$ has the same property $P_k$ for all positive integers $k$.
[i]Proposed by Joshua Brakensiek[/i]
2023 Dutch IMO TST, 1
Find all prime numbers $p$ such that the number
$$3^p+4^p+5^p+9^p-98$$
has at most $6$ positive divisors.
2021 Junior Macedonian Mathematical Olympiad, Problem 4
Let $a$, $b$, $c$ be positive real numbers such that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} = \frac{27}{4}.$ Show that:
$$\frac{a^3+b^2}{a^2+b^2} + \frac{b^3+c^2}{b^2+c^2} + \frac{c^3+a^2}{c^2+a^2} \geq \frac{5}{2}.$$
[i]Authored by Nikola Velov[/i]
2024 Thailand TST, 2
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.
2017 USA Team Selection Test, 3
Let $P, Q \in \mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\lambda$ such that $P + \lambda Q$ is the square of a polynomial.
[i]Alison Miller[/i]
2018 Tajikistan Team Selection Test, 5
Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that:
n|a_i b_i-a_j b_j
[color=#00f]Moved to HSO. ~ oVlad[/color]
2022 JBMO TST - Turkey, 8
Find all prime numbers $p$ such that the number
$$3^p+4^p+5^p+9^p-98$$
has at most $6$ positive divisors.
2020 Vietnam Team Selection Test, 4
Let $n$ be a positive integer. In a $(2n+1)\times (2n+1)$ board, each grid is dyed white or black. In each row and each column, if the number of white grids is smaller than the number of black grids, then we mark all white grids. If the number of white grids is bigger than the number of black grids, then we mark all black grids. Let $a$ be the number of black grids, and $b$ be the number of white grids, $c$ is the number of marked grids.
In this example of $3\times 3$ table, $a=3$, $b=6$, $c=4$. (forget about my watermark)
Proof that no matter how is the dyeing situation in the beginning, there is always $c\geq\frac{1}{2}\min\{a,b\}$.
2004 Germany Team Selection Test, 2
In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$.
Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.