Found problems: 85335
1997 IMO Shortlist, 18
The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$
2021 Bolivian Cono Sur TST, 2
The numbers $1,2,...,100$ are written in a board. We are allowed to choose any two numbers from the board $a,b$ to delete them and replace on the board the number $a+b-1$.
What are the possible numbers u can get after $99$ consecutive operations of these?
2024 India IMOTC, 2
Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that
\[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\]
[i]Proposed by Shantanu Nene[/i]
2024-IMOC, N2
Find all positive integers $(m,n)$ such that
$$11^n+2^n+6=m^3$$
2014 BMT Spring, 2
Regular hexagon $ABCDEF$ has side length $2$ and center $O$. The point $P$ is defined as the intersection of $AC$ and $OB$. Find the area of quadrilateral $OPCD$.
2016 Saudi Arabia Pre-TST, 1.3
Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.
1996 Irish Math Olympiad, 3
A function $ f$ from $ [0,1]$ to $ \mathbb{R}$ has the following properties:
$ (i)$ $ f(1)\equal{}1;$
$ (ii)$ $ f(x) \ge 0$ for all $ x \in [0,1]$;
$ (iii)$ If $ x,y,x\plus{}y \in [0,1]$, then $ f(x\plus{}y) \ge f(x)\plus{}f(y)$.
Prove that $ f(x) \le 2x$ for all $ x \in [0,1]$.
1993 Tournament Of Towns, (389) 1
Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite?
(Folklore)
2006 Romania Team Selection Test, 1
The circle of center $I$ is inscribed in the convex quadrilateral $ABCD$. Let $M$ and $N$ be points on the segments $AI$ and $CI$, respectively, such that $\angle MBN = \frac 12 \angle ABC$. Prove that $\angle MDN = \frac 12 \angle ADC$.
1952 Putnam, B6
Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.
1997 Baltic Way, 13
Five distinct points $A,B,C,D$ and $E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. The point $F$ lies outside the line. Let $G$ be the circumcentre of the triangle $ADF$ and $H$ the circumcentre of the triangle $BEF$. Show that the lines $GH$ and $FC$ are perpendicular.
2023 Stars of Mathematics, 2
Let $a{}$ and $b{}$ be positive integers, whose difference is a prime number. Prove that $(a^n+a+1)(b^n+b+1)$ is not a perfect square for infinitely many positive integers $n{}$.
[i]Proposed by Vlad Matei[/i]
Estonia Open Junior - geometry, 1995.1.2
Two circles of equal radius intersect at two distinct points $A$ and $B$. Let their radii $r$ and their midpoints respectively be $O_1$ and $O_2$. Find the greatest possible value of the area of the rectangle $O_1AO_2B$.
2002 AMC 10, 12
For which of the following values of $ k$ does the equation $ \frac{x\minus{}1}{x\minus{}2}\equal{}\frac{x\minus{}k}{x\minus{}6}$ have no solution for $ x$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
2020 Bulgaria Team Selection Test, 5
Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$.
Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$
2022 Romania Team Selection Test, 1
A finite set $\mathcal{L}$ of coplanar lines, no three of which are concurrent, is called [i]odd[/i] if, for every line $\ell$ in $\mathcal{L}$ the total number of lines in $\mathcal{L}$ crossed by $\ell$ is odd.
[list=a]
[*]Prove that every finite set of coplanar lines, no three of which are concurrent, extends to an odd set of coplanar lines.
[*]Given a positive integer $n$ determine the smallest nonnegative integer $k$ satisfying the following condition: Every set of $n$ coplanar lines, no three of which are concurrent, extends to an odd set of $n+k$ coplanar lines.
[/list]
2022 JHMT HS, 2
Erica intends to construct a subset $T$ of $S=\{ I,J,K,L,M,N \}$, but if she is unsure about including an element $x$ of $S$ in $T$, she will write $x$ in bold and include it in $T$. For example, $\{ I,J \},$ $\{ J,\mathbf{K},L \},$ and $\{ \mathbf{I},\mathbf{J},\mathbf{M},\mathbf{N} \}$ are valid examples of $T$, while $\{ I,J,\mathbf{J},K \}$ is not. Find the total number of such subsets $T$ that Erica can construct.
2018 Azerbaijan BMO TST, 4
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
2013 Vietnam Team Selection Test, 4
Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]
1966 IMO Longlists, 25
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]
2008 Philippine MO, 2
Find the largest integer $n$ for which $\frac{n^{2007}+n^{2006}+\cdots+n^2+n+1}{n+2007}$ is an integer.
2024 Brazil EGMO TST, 4
The infinite sequence \( a_1, a_2, \ldots \) is defined by \( a_1 = 1 \) and, for each \( n \geq 1 \), the number \( a_{n+1} \) is the smallest positive integer greater than \( a_n \) that has the following property: for each \( k \in \{1, 2, \ldots, n\} \), the number \( a_{n+1} + a_k \) is not a perfect square. Prove that, for all \( n \), it holds that \( a_n \leq (n - 1)^2 + 1 \).
2007 AMC 8, 6
The average cost of a long-distance call in the USA in $1985$ was $41$ cents per minute, and the average cost of a long-distance call in the USA in $2005$ was $7$ cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call.
$\textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 17 \qquad
\textbf{(C)}\ 34 \qquad
\textbf{(D)}\ 41 \qquad
\textbf{(E)}\ 80$
2024 Romania EGMO TST, P2
In a park there are 23 trees $t_0,t_1,\dots,t_{22}$ in a circle and 22 birds $b_1,n_2,\dots,b_{22}.$ Initially, each bird is in a tree. Every minute, the bird $b_i, 1\leqslant i\leqslant 22$ flies from the tree $t_j{}$ to the tree $t_{i+j}$ in clockwise order, indices taken modulo 23. Prove that there exists a moment when at least 6 trees are empty.
1994 Tournament Of Towns, (420) 1
Several boys and girls are dancing a waltz at a ball. Is it possible that each girl can always get to dance the next dance with a boy who is either more handsome or more clever than for the previous dance, and that each time one of the girls gets to dance the next dance with a boy who is more handsome and more clever? (The numbers of boys and girls are equal and all are dancing.)
(AY Belov)