Found problems: 85335
2015 Iran Team Selection Test, 5
We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$, $n\nmid a_i+a_k-2a_j$ find all natural numbers $n\ge 3$ such that there exist a "good" permutation of a set $\{1,2,\cdots, n\}$.
2019 Purple Comet Problems, 18
Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying
$a + b - c - d = 3$
$ab - 3bc + cd - 3da = 4$
$3ab - bc + 3cd - da = 5$
Find $11(a - c)^2 + 17(b -d)^2$.
2022 AMC 12/AHSME, 23
Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define
\[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\]
Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum
\[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\]
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 15$
2023 Middle European Mathematical Olympiad, 4
Let $n, m$ be positive integers. A set $S$ of positive integers is called $(n, m)$-good, if:
(1) $m \in S$;
(2) for all $a\in S$, all divisors of $a$ are also in $S$;
(3) for all distinct $a, b \in S$, $a^n+b^n \in S$.
For which $(n, m)$, the only $(n, m)$-good set is $\mathbb{N}$?
2005 Tournament of Towns, 1
A palindrome is a positive integer which reads in the same way in both directions (for example, $1$, $343$ and $2002$ are palindromes, while $2005$ is not). Is it possible to find $2005$ pairs in the form of $(n, n + 110)$ where both numbers are palindromes?
[i](3 points)[/i]
2010 Finnish National High School Mathematics Competition, 5
Let $S$ be a non-empty subset of a plane. We say that the point $P$ can be seen from $A$ if every point from the line segment $AP$ belongs to $S$. Further, the set $S$ can be seen from $A$ if every point of $S$ can be seen from $A$. Suppose that $S$ can be seen from $A$, $B$ and $C$ where $ABC$ is a triangle. Prove that $S$ can also be seen from any other point of the triangle $ABC$.
1992 Brazil National Olympiad, 1
The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$
2010 China National Olympiad, 3
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.
2021 Iran Team Selection Test, 2
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have :
$$f(n)+1400m^2|n^2+f(f(m))$$
2014 Czech-Polish-Slovak Junior Match, 2
Solve the equation $a + b + 4 = 4\sqrt{a\sqrt{b}}$ in real numbers
2023/2024 Tournament of Towns, 1
For every polynomial of degree 45 with coefficients $1,2,3, \ldots, 46$ (in some order) Tom has listed all its distinct real roots. Then he increased each number in the list by 1 . What is now greater: the amount of positive numbers or the amount of negative numbers?
Alexey Glebov
2020 MBMT, 20
Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric?
[i]Proposed by Gabriel Wu[/i]
2016 Indonesia TST, 3
Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$
1994 Austrian-Polish Competition, 6
Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy:
$$\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 \\
(x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 \\
...\\
(x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}$$
Show that there exists $j, 1 \le j \le n$, such that $x_j = x_{j+1}$. Here $x_{n+1} = x_1$.
2024 Korea Summer Program Practice Test, 3
Find all pairs of positive integers $n$ such that one can partition a $n\times (n+1)$ board with $1\times 2$ or $2\times 1$ dominoes and draw one of the diagonals on each of the dominos so that none of the diagonals share endpoints.
2010 Romania National Olympiad, 3
For any integer $n\ge 2$ denote by $A_n$ the set of solutions of the equation
\[x=\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{3}\right\rfloor+\cdots+\left\lfloor\frac{x}{n}\right\rfloor .\]
a) Determine the set $A_2\cup A_3$.
b) Prove that the set $A=\bigcup_{n\ge 2}A_n$ is finite and find $\max A$.
[i]Dan Nedeianu & Mihai Baluna[/i]
2010 Princeton University Math Competition, 5
We say that a rook is "attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let $n$ be the maximum number of rooks that can be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other. How many ways can $n$ rooks be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other?
2019 India IMO Training Camp, P3
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.
2010 China Team Selection Test, 2
Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set
$A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$
$B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$
Prove that $2|A|\geq |B|$.
Novosibirsk Oral Geo Oly IX, 2016.1
In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.
2007 Princeton University Math Competition, 4
Find all values of $a$ such that $x^6 - 6x^5 + 12x^4 + ax^3 + 12x^2 - 6x +1$ is nonnegative for all real $x$.
2007 Romania Team Selection Test, 3
Let $a_{i}$, $i = 1,2, \dots ,n$, $n \geq 3$, be positive integers, having the greatest common divisor 1, such that \[a_{j}\textrm{ divide }\sum_{i = 1}^{n}a_{i}\]
for all $j = 1,2, \dots ,n$. Prove that \[\prod_{i = 1}^{n}a_{i}\textrm{ divides }\Big{(}\sum_{i = 1}^{n}a_{i}\Big{)}^{n-2}.\]
1993 China Team Selection Test, 2
Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$
2001 Chile National Olympiad, 1
$\bullet$ In how many ways can triangles be formed whose sides are integers greater than $50$ and less than $100$?
$\bullet$ In how many of these triangles is the perimeter divisible by $3$?
2007 Indonesia TST, 3
Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.