Found problems: 85335
1970 Canada National Olympiad, 8
Consider all line segments of length 4 with one end-point on the line $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.
PEN O Problems, 55
The set $M$ consists of integers, the smallest of which is $1$ and the greatest $100$. Each member of $M$, except $1$, is the sum of two (possibly identical) numbers in $M$. Of all such sets, find one with the smallest possible number of elements.
2015 Postal Coaching, Problem 5
Let $ABCD$ be a convex quadrilateral. In the triangle $ABC$ let $I$ and $J$ be the incenter and the excenter opposite to vertex $A$, respectively. In the triangle $ACD$ let $K$ and $L$ be the incenter and the excenter opposite to vertex $A$, respectively. Show that the lines $IL$ and $JK$, and the bisector of the angle $BCD$ are concurrent.
1992 IMO Longlists, 35
Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.
1967 IMO Shortlist, 4
Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that:
a.) Using medians of that triangle it is possible to construct a rectangular triangle.
b.) The following inequality: \[5(a^2+b^2-c^2) \geq 8ab,\] is valid, where $a,b$ and $c$ are side length of the given triangle.
2011 Dutch IMO TST, 3
The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.
1990 Irish Math Olympiad, 4
Let $n=2k-1$, where $k\ge 6$ is an integer. Let $T$ be the set of all $n$-tuples $$\textbf{x}=(x_1,x_2,\dots ,x_n), \text{ where, for } i=1,2,\dots ,n, \text{ } x_i \text{ is } 0 \text{ or } 1.$$ For $\textbf{x}=(x_1,x_2,\dots ,x_n)$ and $\textbf{y}=(y_1,y_2,\dots ,y_n)$ in $T$, let $d(\textbf{x},\textbf{y})$ denote the number of integers $j$ with $1\le j\le n$ such that $x_j\neq x_y$. $($In particular, $d(\textbf{x},\textbf{x})=0)$.
Suppose that there exists a subset $S$ of $T$ with $2^k$ elements which has the following property: given any element $\textbf{x}$ in $T$, there is a unique $\textbf{y}$ in $S$ with $d(\textbf{x},\textbf{y})\le 3$.
Prove that $n=23$.
2003 IMC, 2
Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)
2019 Durer Math Competition Finals, 12
How many ways are there to arrange the numbers $1, 2, 3, .. , 15$ in some order such that for any two numbers which are $2$ or $3$ positions apart, the one on the left is greater?
2009 Saint Petersburg Mathematical Olympiad, 7
Points $Y,X$ lies on $AB,BC$ of $\triangle ABC$ and $X,Y,A,C$ are concyclic. $AX$ and $CY$ intersect in $O$. Points $M,N$ are midpoints of $AC$ and $XY$. Prove, that $BO$ is tangent to circumcircle of $\triangle MON$