Found problems: 85335
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.
1959 Miklós Schweitzer, 4
[b]4.[/b] Consider $n$ circles of radius $1$ in the planea. Prove that at least one of the circles contains an are of length greater than $\frac{2\pi}{n}$ not intersected by any other of these circles. [b](G. 4)[/b]
2010 Malaysia National Olympiad, 6
A two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.
2007 Moldova National Olympiad, 10.5
In a chess tournament , each of two players have only one game played. After 2 rounds 5 players left the tournament. At the final of tournament was found that the number of total games played is 100. How many players were at the start of the tournament?
1974 Yugoslav Team Selection Test, Problem 2
Given two directly congruent triangles $ABC$ and $A'B'C'$ in a plane, assume that the circles with centers $C$ and $C'$ and radii $CA$ and $C'A'$ intersect. Denote by $\mathcal M$ the transformation that maps $\triangle ABC$ to $\triangle A'B'C'$. Prove that $\mathcal M$ can be expressed as a composition of at most three rotations in the following way: The first rotation has the center in one of $A,B,C$ and maps $\triangle ABC$ to $\triangle A_1B_1C_1$; The second rotation has the center in one of $A_1,B_1,C_1$, and maps $\triangle A_1B_1C_1$ to $\triangle A_2B_2C_2$; The third rotation has the center in one of $A_2,B_2,C_2$ and maps $\triangle A_2B_2C_2$ to $\triangle A'B'C'$.
2004 IberoAmerican, 3
Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$
2016 MMPC, 5
Consider four real numbers $x$, $y$, $a$, and $b$, satisfying $x + y = a + b$ and $x^2 + y^2 = a^2 + b^2$. Prove that $x^n + y^n = a^n + b^n$, for all $n \in \mathbb{N}$.
2025 District Olympiad, P4
Let $ABCDEF$ be a convex hexagon with $\angle A = \angle C=\angle E$ and $\angle B = \angle D=\angle F$.
[list=a]
[*] Prove that there is a unique point $P$ which is equidistant from sides $AB,CD$ and $EF$.
[*] If $G_1$ and $G_2$ are the centers of mass of $\triangle ACE$ and $\triangle BDF$, show that $\angle G_1PG_2=60^{\circ}$.
JBMO Geometry Collection, 1997
Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$.
[i]Greece[/i]
2003 Gheorghe Vranceanu, 2
Let $ a $ be a positive real number and $ \left( x_n\right)_{n\ge 1} $ be a sequence of pairwise distinct real numbers satisfying the properties:
$ \text{(i) } x_n\in (0,a) , $ for any natural numbers $ n $
$ \text{(ii) } \left| x_n-x_m \right|\geqslant\frac{m+n}{amn} , $ for all pairs $ (m,n) $ of distinct natural numbers
Show that $ a\geqslant 2. $
1987 Tournament Of Towns, (142) 2
In $3$ dimensional space we are given a parallelogram $ABCD$ and plane $M$. The distances from vertices $A, B$ and $C$ to plane $M$ are $a, b$ and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$ .