Found problems: 85335
2008 Oral Moscow Geometry Olympiad, 1
Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar?
(D. Shnol)
1996 German National Olympiad, 6b
Each point of a plane is colored in one of three colors: red, black and blue. Prove that there exists a rectangle in this plane whose vertices all have the same color.
2012 Sharygin Geometry Olympiad, 5
A quadrilateral $ABCD$ with perpendicular diagonals is inscribed into a circle $\omega$. Two arcs $\alpha$ and $\beta$ with diameters AB and $CD$ lie outside $\omega$. Consider two crescents formed by the circle $\omega$ and the arcs $\alpha$ and $\beta$ (see Figure). Prove that the maximal radii of the circles inscribed into these crescents are equal.
(F.Nilov)
2013 Romania Team Selection Test, 3
Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.
2013 China National Olympiad, 2
For any positive integer $n$ and $0 \leqslant i \leqslant n$, denote $C_n^i \equiv c(n,i)\pmod{2}$, where $c(n,i) \in \left\{ {0,1} \right\}$. Define
\[f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}\]
where $m,n,q$ are positive integers and $q + 1 \ne {2^\alpha }$ for any $\alpha \in \mathbb N$. Prove that if $f(m,q)\left| {f(n,q)} \right.$, then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$.
2009 Tuymaada Olympiad, 3
On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$?
[i]Proposed by S. Berlov[/i]
1973 AMC 12/AHSME, 22
The set of all real solutions of the inequality
\[ |x \minus{} 1| \plus{} |x \plus{} 2| < 3\]
is
$ \textbf{(A)}\ x \in ( \minus{} 3,2) \qquad \textbf{(B)}\ x \in ( \minus{} 1,2) \qquad \textbf{(C)}\ x \in ( \minus{} 2,1) \qquad$
$ \textbf{(D)}\ x \in \left( \minus{} \frac32,\frac72\right) \qquad \textbf{(E)}\ \O \text{ (empty})$
Note: I updated the notation on this problem.
Kyiv City MO Juniors 2003+ geometry, 2010.89.4
Point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AO$ intersects the side $BC$ at point $D$ so that $OD = BD = 1/3 BC$ . Find the angles of the triangle $ABC$. Justify the answer.
2017 CMIMC Individual Finals, 2
Kevin likes drawing. He takes a large piece of paper and draws on it every rectangle with positive integer side lengths and perimeter at most 2017, with no two rectangles overlapping. Compute the total area of the paper that is covered by a rectangle.
2023 Greece Junior Math Olympiad, 3
Find the number of rectangles who have the following properties:
a) Have for vertices, points $(x,y)$ of plane $Oxy$ with $x,y$ non negative integers and $ x \le 8$ , $y\le 8$
b) Have sides parallel to axes
c) Have area $E$, with $30<E\le 40$
III Soros Olympiad 1996 - 97 (Russia), 10.4
Find natural $a, b, c, d$ that satisfy the system
$$\begin{cases} ab+cd=34
\\ ac-bd=19
\end{cases}$$
2015 District Olympiad, 3
Find all continuous and nondecreasing functions $ f:[0,\infty)\longrightarrow\mathbb{R} $ that satisfy the inequality:
$$ \int_0^{x+y} f(t) dt\le \int_0^x f(t) dt +\int_0^y f(t) dt,\quad\forall x,y\in [0,\infty) . $$
2003 Poland - Second Round, 5
Point $A$ lies outside circle $o$ of center $O$. From point $A$ draw two lines tangent to a circle $o$ in points $B$ and $C$. A tangent to a circle $o$ cuts segments $AB$ and $AC$ in points $E$ and $F$, respectively. Lines $OE$ and $OF$ cut segment $BC$ in points $P$ and $Q$, respectively. Prove that from line segments $BP$, $PQ$, $QC$ can construct triangle similar to triangle $AEF$.
2014 Costa Rica - Final Round, 5
Let $ABC$ be a triangle, with $A'$, $B'$, and $C'$ the points of tangency of the incircle with $BC$, $CA$, and $AB$ respectively. Let $X$ be the intersection of the excircle with respect to $A$ with $AB$, and $M$ the midpoint of $BC$. Let $D$ be the intersection of $XM$ with $B'C'$. Show that $\angle C'A'D' = 90^o$.
1990 Polish MO Finals, 3
Prove that for all integers $n > 2$,
\[ 3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i} \]
2018 JHMT, 1
Let $m$ be the area and let $n$ be the perimeter of a regular octagon. The ratio $\frac{m^2}{n}$ can be expressed as $p \tan (q \pi)$ where $p$ is a positive integer. Find $pq$.
1952 AMC 12/AHSME, 37
Two equal parallel chords are drawn $ 8$ inches apart in a circle of radius $ 8$ inches. The area of that part of the circle that lies between the chords is:
$ \textbf{(A)}\ 21\frac {1}{3}\pi \minus{} 32\sqrt {3} \qquad\textbf{(B)}\ 32\sqrt {3} \plus{} 21\frac {1}{3}\pi \qquad\textbf{(C)}\ 32\sqrt {3} \plus{} 42\frac {2}{3}\pi$
$ \textbf{(D)}\ 16\sqrt {3} \plus{} 42\frac {2}{3}\pi \qquad\textbf{(E)}\ 42\frac {2}{3}\pi$
2011 Bogdan Stan, 1
Find the natural numbers $ n $ which have the property that
$$ 2011=\left| \mathbb{Q}\cap\bigcup_{k=1}^n\left\{ x\in\mathbb{R} | 1+k^2x^2=2k\left( x-\lfloor x\rfloor \right) \right\} \right| . $$
[i]Marian Teler[/i]
2004 Abels Math Contest (Norwegian MO), 1b
Let $a_1,a_2,a_3,...$ be a strictly increasing sequence of positive integers. A number $a_n$ in the sequence is said to be [i]lucky [/i] if it is the sum of several (not necessarily distinct) smaller terms of the sequence, and [i]unlucky [/i]otherwise. (For example, in the sequence $4,6,14,15,25,...$ numbers $4,6,15$ are [i]unlucky[/i], while $14 = 4+4+6$ and $25 = 4+6+15$ are [i]lucky[/i].) Prove that there are only finitely many [i]unlucky [/i]numbers in the sequence.
2011 Philippine MO, 5
The chromatic number $\chi$ of an (infinite) plane is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart.
Prove that $4 \leq \chi \leq 7$.
2014 JHMMC 7 Contest, 22
For how many positive integer values of $x$ is $4^x- 1$ prime?
2018 Harvard-MIT Mathematics Tournament, 7
Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q\cdot 15!$ for some rational number $q$. Find $q$.
1953 Putnam, B7
Let $w\in (0,1)$ be an irrational number. Prove that $w$ has a unique convergent expansion of the form
$$w= \frac{1}{p_0} - \frac{1}{p_0 p_1 } + \frac{1}{ p_0 p_1 p_2 } - \frac{1}{p_0 p_1 p_2 p_3 } +\ldots,$$
where $1\leq p_0 < p_1 < p_2 <\ldots $ are integers. If $w= \frac{1}{\sqrt{2}},$ find $p_0 , p_1 , p_2.$
2015 India PRMO, 9
$9.$ What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12 ?$
2019 Serbia JBMO TST, 3
$3.$ Congruent circles $k_{1}$ and $k_{2}$ intersect in the points $A$ and $B$. Let $P$ be a variable point of arc $AB$ of circle $k_{2}$ which is inside $k_{1}$ and let $AP$ intersect $k_{1}$ once more in point $C$, and the ray $CB$ intersects $k_{2}$ once more in $D$. Let the angle bisector of $\angle CAD$ intersect $k_{1}$ in $E$, and the circle $k_{2}$ in $F$. Ray $FB$ intersects $k_{1}$ in $Q$. If $X$ is one of the intersection points of circumscribed circles of triangles $CDP$ and $EQF$, prove that the triangle $CFX$ is equilateral.