Found problems: 85335
2006 Belarusian National Olympiad, 5
A convex quadrilateral $ABCD$ Is placed on the Cartesian plane. Its vertices $A$ and $D$ belong to the negative branch of the graph of the hyperbola $y= 1/x$, the vertices $B$ and $C$ belong to the positive branch of the graph and point $B$ lies at the left of $C$, the segment $AC$ passes through the origin $(0,0)$. Prove that $\angle BAD = \angle BCD$.
(I, Voronovich)
2020 Sharygin Geometry Olympiad, 20
The line touching the incircle of triangle $ABC$ and parallel to $BC$ meets the external bisector of angle $A$ at point $X$. Let $Y$ be the midpoint of arc $BAC$ of the circumcircle. Prove that the angle $XIY$ is right.
1996 Swedish Mathematical Competition, 3
For every positive integer $n$, we define the function $p_n$ for $x\ge 1$ by
$$p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).$$
Prove that $p_n(x) \ge 1$ and that $p_{mn}(x) = p_m(p_n(x))$.
2014 Iran Geometry Olympiad (senior), 5:
Two points $P$ and $Q$ lying on side $BC$ of triangle $ABC$ and their distance from the midpoint of $BC$ are equal.The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$,respectively.$M$ is point of intersection $PF$ and $EQ$.If $H_1$ and $H_2$ be the orthocenters of triangles $BFP$ and $CEQ$, respectively, prove that $ AM\perp H_1H_2 $.
Author:Mehdi E'tesami Fard , Iran
2019 New Zealand MO, 7
Let $ABCDEF$ be a convex hexagon containing a point $P$ in its interior such that $PABC$ and $PDEF$ are congruent rectangles with $PA = BC = P D = EF$ (and $AB = PC = DE = PF$). Let $\ell$ be the line through the midpoint of $AF$ and the circumcentre of $PCD$. Prove that $\ell$ passes through $P$.
2016 District Olympiad, 2
Show that:
$$ 2015\in\left\{ x_1+2x_2+3x_3\cdots +2015x_{2015}\big| x_1,x_2,\ldots ,x_{2015}\in \{ -2,3\}\right\}\not\ni 2016. $$
2004 AMC 10, 6
Which of the following numbers is a perfect square?
$ \textbf{(A)}\ 98!\cdot 99!\qquad \textbf{(B)}\ 98!\cdot 100!\qquad \textbf{(C)}\ 99!\cdot 100!\qquad \textbf{(D)}\ 99!\cdot 101!\qquad \textbf{(E)}\ 100!\cdot 101!$
2010 Princeton University Math Competition, 7
Find the numerator of \[\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\] when reduced.
2018 China Team Selection Test, 4
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.
2019 Romania National Olympiad, 3
$\textbf{a)}$ Prove that there exists a differentiable function $f:(0, \infty) \to (0, \infty)$ such that $f(f'(x)) = x, \: \forall x>0.$
$\textbf{b)}$ Prove that there is no differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$
1999 AIME Problems, 15
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
Indonesia MO Shortlist - geometry, g5
Two circles intersect at points $A$ and $B$. The line $\ell$ through A intersects the circles at $C$ and $D$, respectively. Let $M, N$ be the midpoints of arc $BC$ and arc $BD$. which does not contain $A$, and suppose that $K$ is the midpoint of the segment $CD$ . Prove that $\angle MKN=90^o$.
2021-IMOC, G11
The incircle of $\triangle ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The projections of $B$, $C$ to $AD$ are $U$, $V$, respectively; the projections of $C$, $A$ to $BE$ are $W$, $X$, respectively; and the projections of $A$, $B$ to $CF$ are $Y$, $Z$, respectively. Show that the circumcircle of the triangle formed by $UX$, $VY$, $WZ$ is tangent to the incircle of $\triangle ABC$.
2008 AMC 10, 13
For each positive integer $ n$, the mean of the first $ n$ terms of a sequence is $ n$. What is the $ 2008$th term of the sequence?
$ \textbf{(A)}\ 2008 \qquad
\textbf{(B)}\ 4015 \qquad
\textbf{(C)}\ 4016 \qquad
\textbf{(D)}\ 4,030,056 \qquad
\textbf{(E)}\ 4,032,064$
2015 Bundeswettbewerb Mathematik Germany, 3
Let $M$ be the midpoint of segment $[AB]$ in triangle $\triangle ABC$. Let $X$ and $Y$ be points such that $\angle{BAX}=\angle{ACM}$ and $\angle{BYA}=\angle{MCB}$. Both points, $X$ and $Y$, are on the same side as $C$ with respect to line $AB$.
Show that the rays $[AX$ and $[BY$ intersect on line $CM$.
2022 AMC 10, 25
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$
1981 Miklós Schweitzer, 4
Let $ G$ be finite group and $ \mathcal{K}$ a conjugacy class of $ G$ that generates $ G$. Prove that the following two statements are equivalent:
(1) There exists a positive integer $ m$ such that every element of $ G$ can be written as a product of $ m$ (not necessarily distinct) elements of $ \mathcal{K}$.
(2) $ G$ is equal to its own commutator subgroup.
[i]J. Denes[/i]
1999 National High School Mathematics League, 2
The number of intengral points $(x,y)$ that fit $(|x|-1)^2+(|y|-1)^2<2$ is
$\text{(A)}16\qquad\text{(B)}17\qquad\text{(C)}18\qquad\text{(D)}25$
2024 JHMT HS, 1
Compute the smallest positive integer $N$ for which $N \cdot 2^{2024}$ is a multiple of $2024$.
2006 Estonia Math Open Junior Contests, 10
Let a, b, c be positive integers. Prove that the inequality
\[ (x\minus{}y)^a(x\minus{}z)^b(y\minus{}z)^c \ge 0
\]
holds for all reals x, y, z if and only if a, b, c are even.
2013 MTRP Senior, 4
Let n be an integer such that if d | n then d + 1 | n + 1. Show that n is a prime number.
1995 National High School Mathematics League, 12
Set $M=\{1,2,\cdots,1995\}$. $A$ is a subset of $M$ such that $\forall x\in A,15x\not\in A$. Then the maximum $|A|$ is________.
2018 JHMT, 4
Equilateral triangle $OAB$ of side length $1$ lies in the $xy$-plane ($O$ is the origin). Let $\ell, m$ be the vertical lines passing through $A,B$, respectively. Let $P,Q$ be on $\ell, m$ respectively such that the ratio $\overline{OP} : \overline{OQ} : \overline{PQ} = 3 : 3 : 5$. Let $Q = (x, y, z)$. If $z^2 = \frac{p}{q}$ . where $p, q$ are relatively prime positive integers, find $p + q$.
2017 IFYM, Sozopol, 6
Find all functions $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$, for which
$f(k+1)>f(f(k)) \quad \forall k \geq 1$.
PEN L Problems, 4
The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $F_{mn}-F_{n+1}^{m}+F_{n-1}^{m}$ is divisible by $F_{n}^{3}$ for all $m \ge 1$ and $n>1$.