Found problems: 85335
1993 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB = 8$ and $CD = 6$, find the distance between the midpoints of $AD$ and $BC$.
2021 Turkey MO (2nd round), 6
In a school, there are 2021 students, each having exactly $k$ friends. There aren't three students such that all three are friends with each other. What is the maximum possible value of $k$?
2017 Junior Regional Olympiad - FBH, 2
In quadrilateral $ABCD$ holds $AB=6$, $AD=4$, $\angle DAB=\angle ABC = 60^{\circ}$ and $\angle ADC = 90^{\circ}$. Find length of diagonals and area of the quadrilateral
2005 Greece Team Selection Test, 3
Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.
2013 Kosovo National Mathematical Olympiad, 2
Three numbers have sum $k$ (where $k\in \mathbb{R}$) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add $\frac{k}{6}$ than we have geometry progression.
Find those numbers?
2021 Auckland Mathematical Olympiad, 3
For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number?
2019 Belarus Team Selection Test, 4.1
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
[list=1]
[*] each triangle from $T$ is inscribed in $\omega$;
[*] no two triangles from $T$ have a common interior point.
[/list]
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
1992 Mexico National Olympiad, 1
The tetrahedron $OPQR$ has the $\angle POQ = \angle POR = \angle QOR = 90^o$. $X, Y, Z$ are the midpoints of $PQ, QR$ and $RP.$ Show that the four faces of the tetrahedron $OXYZ$ have equal area.
2011 Morocco National Olympiad, 3
Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.
2023 Durer Math Competition Finals, 5
For an acute triangle $ABC$, let $O$ be its circumcenter, and let $O_A,O_B,O_C$ be the circumcenter of $BCO,CAO,ABO$ respectively. Show that $AO_A,BO_B,CO_C$ are concurrent.
2019 Belarus Team Selection Test, 6.3
Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence
\[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \]
forms an arithmetic progression. Prove that the terms of the sequence are equal.
2017 India IMO Training Camp, 3
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively.
(a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic.
(b) If $R$ is the radius of $\Omega$ and $d$ is the distance between its centre and $P$, then find the radius of the circle in (a) in terms of $R$ and $d$.
2009 Czech-Polish-Slovak Match, 1
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy \[ \Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1\] for all $x,y\in\mathbb{R}^+$.
Kyiv City MO Seniors Round2 2010+ geometry, 2013.11.4
Let $ H $ be the intersection point of the altitudes $ AP $ and $ CQ $ of the acute-angled triangle $ ABC $. On its median $ BM $ marked points $ E $ and $ F $ so that $ \angle APE = \angle BAC $ and $ \angle CQF = \angle BCA $, and the point $ E $ lies inside the triangle $ APB $, and the point $ F $ lies inside the triangle $ CQB $. Prove that the lines $ AE $, $ CF $ and $ BH $ intersect at one point.
(Vyacheslav Yasinsky)
2021 BMT, 5
Anthony the ant is at point $A$ of regular tetrahedron $ABCD$ with side length $4$. Anthony wishes to crawl on the surface of the tetrahedron to the midpoint of $\overline{BC}$. However, he does not want to touch the interior of face $\vartriangle ABC$, since it is covered with lava. What is the shortest distance Anthony must travel?
2014 JHMMC 7 Contest, 23
An isosceles triangle has side lengths $x-4, 2x -9,\text{and}3x - 15$. Find the sum of all possible values of $x$.
2024-25 IOQM India, 15
Let $X$ be the set of consisting of twenty positive integers $n,n+2,...,n+38$. The smallest value of $n$ for which any three numbers $a,b,c \in X$, not necessarily distinct, form the sides of an acute-angled triangle is:
IV Soros Olympiad 1997 - 98 (Russia), 10.7
Prove that the number $\left(\sqrt2+\sqrt3+\sqrt5\right)^{1997}$ can be represented as $$A\sqrt2+B\sqrt3+C\sqrt5+D\sqrt{30}$$ where $A$, $B$, $C$, $D$ are integers. Find with approximation to $10^{-10}$ the ratio $\frac{D}{A}$
2002 District Olympiad, 2
A group of $67$ students pass their examination consisting of $6$ questions, labeled with the numbers $1$ to $6$. A correct answer to question $n$ is quoted $n$ points and for an incorrect answer to the same question a student loses $n$ point.
a) Find the least possible positive difference between any $2$ final scores
b) Show that at least $4$ participants have the same final score
c) Show that at least $2$ students gave identical answer to all six questions.
1996 China Team Selection Test, 2
Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that
\[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\]
Define
\begin{align*}
A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\
B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\
W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2.
\end{align*}
Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.
1965 AMC 12/AHSME, 31
The number of real values of $ x$ satisfying the equality $ (\log_2x)(\log_bx) \equal{} \log_ab$, where $ a > 0$, $ b > 0$, $ a \neq 1$, $ b \neq 1$, is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{a finite integer greater than 2} \qquad \textbf{(E)}\ \text{not finite}$
2008 Mathcenter Contest, 2
In triangle $ABC$ ($AB\not= AC$), the incircle is tangent to the sides of $BC$ ,$CA$ , $AB$ at $D$ ,$E$, $F$ respectively. Let $AD$ meet the incircle again at point $P$, let $EF$ and the line passing through the point $P$ and perpendicular to $AD$ intersect at $Q$. Let $AQ$ intersect $DE$ at $X$ and $DF$ at $Y$. Prove that $AX=AY$.
[i](tatari/nightmare)[/i]
1936 Moscow Mathematical Olympiad, 029
The lengths of a rectangle’s sides and of its diagonal are integers. Prove that the area of the rectangle is an integer multiple of $12$.
1989 India National Olympiad, 4
Determine all $n \in \mathbb{N}$ for which [list][*] $n$ is not the square of any integer, [*] $\lfloor \sqrt{n}\rfloor ^3$ divides $n^2$. [/list]
1951 AMC 12/AHSME, 26
In the equation $ \frac {x(x \minus{} 1) \minus{} (m \plus{} 1)}{(x \minus{} 1)(m \minus{} 1)} \equal{} \frac {x}{m}$ the roots are equal when
$ \textbf{(A)}\ m \equal{} 1 \qquad\textbf{(B)}\ m \equal{} \frac {1}{2} \qquad\textbf{(C)}\ m \equal{} 0 \qquad\textbf{(D)}\ m \equal{} \minus{} 1 \qquad\textbf{(E)}\ m \equal{} \minus{} \frac {1}{2}$