Found problems: 85335
2014 Olympic Revenge, 4
Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that
\[n \mid a^{f(n)}-1.\]
Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.
1994 Vietnam Team Selection Test, 1
Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$.
[b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$.
[b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.
2008 Bosnia And Herzegovina - Regional Olympiad, 3
A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?
2024 Iran MO (3rd Round), 3
$m,n$ are given integer numbers such that $m+n$ is an odd number. Edges of a complete bipartie graph $K_{m,n}$ are labeled by ${-1,1}$ such that the sum of all labels is $0$. Prove that there exists a spanning tree such that the sum of the labels of its edges is equal to $0$.
1942 Putnam, A4
Find the orthogonal trajectories of the family of conics $(x+2y)^{2} = a(x+y)$. At what angle do the curves of one family cut the curves of the other family at the origin?
1991 IMO Shortlist, 16
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
2007 National Olympiad First Round, 5
Let $C$ and $D$ be points on the semicircle with center $O$ and diameter $AB$ such that $ABCD$ is a convex quadrilateral. Let $Q$ be the intersection of the diagonals $[AC]$ and $[BD]$, and $P$ be the intersection of the lines tangent to the semicircle at $C$ and $D$. If $m(\widehat{AQB})=2m(\widehat{COD})$ and $|AB|=2$, then what is $|PO|$?
$
\textbf{(A)}\ \sqrt 2
\qquad\textbf{(B)}\ \sqrt 3
\qquad\textbf{(C)}\ \frac{1+\sqrt 3} 2
\qquad\textbf{(D)}\ \frac{1+\sqrt 3}{2\sqrt 2}
\qquad\textbf{(E)}\ \frac{2\sqrt 3} 3
$
2018 Cyprus IMO TST, 2
Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.
2008 Bosnia And Herzegovina - Regional Olympiad, 3
Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.
2016 Hanoi Open Mathematics Competitions, 15
Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$.
Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.
2022 Ecuador NMO (OMEC), 4
Find the number of sets with $10$ distinct positive integers such that the average of its elements is less than 6.
2019 Brazil National Olympiad, 5
(a) Prove that given constants $a,b$ with $1<a<2<b$, there is no partition of the set of positive integers into two subsets $A_0$ and $A_1$ such that: if $j \in \{0,1\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.
(b) Find all pairs of real numbers $(a,b)$ with $1<a<2<b$ for which the following property holds: there exists a partition of the set of positive integers into three subsets $A_0, A_1, A_2$ such that if $j \in \{0,1,2\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.
2012 IMO Shortlist, A2
Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively.
a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.
1985 Spain Mathematical Olympiad, 8
A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant.
Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.
2020 Ecuador NMO (OMEC), 2
Find all pairs $(n, q)$ such that $n$ is a positive integer, $q$ is a not integer rational and
$$n^q-q$$
is an integer.
2017 ELMO Shortlist, 3
Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs.
[i]Proposed by Daniel Liu
2007 Harvard-MIT Mathematics Tournament, 4
Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$.
2025 Caucasus Mathematical Olympiad, 5
Given a $20 \times 25$ board whose rows are numbered from $1$ to $20$ and whose columns are numbered from $1$ to $25$, Nikita wishes to place one precious stone in some cells of this board so that at least one stone is present and the following magical condition holds: for any $1 \leqslant i \leqslant 20$ and $1 \leqslant j \leqslant 25$, there is a stone in the cell at the intersection of the $i^\text{th}$ row and the $j^\text{th}$ column if and only if the cross formed by the union of the $i^\text{th}$ row and the $j^\text{th}$ column contains exactly $i + j$ stones. Determine whether Nikita's wish is achievable.
2010 Saudi Arabia Pre-TST, 3.2
Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.
2023 Saint Petersburg Mathematical Olympiad, 7
Let $\ell_1, \ell_2$ be two non-parallel lines and $d_1, d_2$ be positive reals. The set of points $X$, such that $dist(X, \ell_i)$ is a multiple of $d_i$ is called a $\textit{grid}$. Let $A$ be finite set of points, not all collinear. A triangle with vertices in $A$ is called $\textit{empty}$ if no points from $A$ lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that $A$ is the intersection of a grid $L$ and a convex polygon $F$.
2024 Kyiv City MO Round 2, Problem 2
Mykhailo wants to arrange all positive integers from $1$ to $2024$ in a circle so that each number is used exactly once and for any three consecutive numbers $a, b, c$ the number $a + c$ is divisible by $b + 1$. Can he do it?
[i]Proposed by Fedir Yudin[/i]
2012 Online Math Open Problems, 17
Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation
\[x^2(x^2+2z) - y^2(y^2+2z)=a.\]
[i]Ray Li.[/i]
[hide="Clarifications"][list=1][*]$x,y,z$ are not necessarily distinct.[/list][/hide]
2018 India IMO Training Camp, 3
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.
2007 AMC 8, 16
Amanda Reckonwith draws five circles with radii 1, 2, 3, 4 and 5. Then for each circle she plots the point (C; A), where C is its circumference and A is its area. Which of the following could be her graph?
$\textbf{(A)}$
[asy]
size(75);
pair A= (1.5,1) ,
B= (3,3) ,
C= (4.5,6) ,
D= (6,10) ,
E= (7.5,15) ;
draw((0,-1)--(0,16));
draw((-1,0)--(16,0));
dot(A^^B^^C^^D^^E);
label("$A$", (0,8), W);
label("$C$", (8,0), S);[/asy]
$\textbf{(B)}$
[asy]
size(75);
pair A= (1.5,9) ,
B= (3,6) ,
C= (4.5,6) ,
D= (6,9) ,
E= (7.5,15) ;
draw((0,-1)--(0,16));
draw((-1,0)--(16,0));
dot(A^^B^^C^^D^^E);
label("$A$", (0,8), W);
label("$C$", (8,0), S);[/asy]
$\textbf{(C)}$
[asy]
size(75);
pair A= (1.5,2) ,
B= (3,6) ,
C= (4.5,8) ,
D= (6,6) ,
E= (7.5,2) ;
draw((0,-1)--(0,16));
draw((-1,0)--(16,0));
dot(A^^B^^C^^D^^E);
label("$A$", (0,8), W);
label("$C$", (8,0), S);[/asy]
$\textbf{(D)}$
[asy]
size(75);
pair A= (1.5,2) ,
B= (3,5) ,
C= (4.5,8) ,
D= (6,11) ,
E= (7.5,14) ;
draw((0,-1)--(0,16));
draw((-1,0)--(16,0));
dot(A^^B^^C^^D^^E);
label("$A$", (0,8), W);
label("$C$", (8,0), S);[/asy]
$\textbf{(E)}$
[asy]
size(75);
pair A= (1.5,15) ,
B= (3,10) ,
C= (4.5,6) ,
D= (6,3) ,
E= (7.5,1) ;
draw((0,-1)--(0,16));
draw((-1,0)--(16,0));
dot(A^^B^^C^^D^^E);
label("$A$", (0,8), W);
label("$C$", (8,0), S);[/asy]
2000 Belarus Team Selection Test, 2.2
Real numbers $a$, $b$, $c$ satisfy the equation $$2a^3-b^3+2c^3-6a^2b+3ab^2-3ac^2-3bc^2+6abc=0$$. If $a<b$, find which of the numbers $b$, $c$ is larger.