Found problems: 85335
2024 Princeton University Math Competition, A1 / B3
The following three squares are inscribed within each other such that they all share the same center, and the largest and smallest squares have parallel sides. If the largest square has side length $17$ and the middle square has side length $13,$ the side length of the smallest square can be expressed in the form $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a + b.$
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Novosibirsk Oral Geo Oly IX, 2019.4
Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.
2015 Albania JBMO TST, 4
For every positive integer $n{}$, consider the numbers $a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46.$
a) Prove that $a_1+a_2+a_3$ is even.
b) Find all positive integers $n$ for which $a_1, a_2$ and $a_3$ are primes.
2010 ELMO Shortlist, 2
Let $a,b,c$ be positive reals. Prove that
\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]
[i]Calvin Deng.[/i]
2008 Sharygin Geometry Olympiad, 1
(B.Frenkin) An inscribed and circumscribed $ n$-gon is divided by some line into two inscribed and circumscribed polygons with different numbers of sides. Find $ n$.
1970 Swedish Mathematical Competition, 6
Show that $\frac{(n - m)!}{m!} \le \left(\frac{n}{2} + \frac{1}{2}\right)^{n-2m}$ for positive integers $m, n$ with $2m \le n$.
1992 Polish MO Finals, 1
Segments $AC$ and $BD$ meet at $P$, and $|PA| = |PD|$, $|PB| = |PC|$. $O$ is the circumcenter of the triangle $PAB$. Show that $OP$ and $CD$ are perpendicular.
2013 JBMO TST - Turkey, 7
In a convex quadrilateral $ABCD$ diagonals intersect at $E$ and $BE = \sqrt{2}\cdot ED, \: \angle BEC = 45^{\circ}.$ Let $F$ be the foot of the perpendicular from $A$ to $BC$ and $P$ be the second intersection point of the circumcircle of triangle $BFD$ and line segment $DC$. Find $\angle APD$.
2019 International Zhautykov OIympiad, 5
Natural number $n>1$ is given. Let $I$ be a set of integers that are relatively prime to $n$. Define the function $f:I=>N$. We call a function $k-periodic$ if for any $a,b$ , $f(a)=f(b)$ whenever $ k|a-b $. We know that $f$ is $n-periodic$. Prove that minimal period of $f$ divides all other periods.
Example: if $n=6$ and $f(1)=f(5)$ then minimal period is 1, if $f(1)$ is not equal to $f(5)$ then minimal period is 3.
2004 Italy TST, 1
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$
2014 IMO Shortlist, G3
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$.
(Here we always assume that an angle bisector is a ray.)
[i]Proposed by Sergey Berlov, Russia[/i]
2014 India PRMO, 19
Let $x_1,x_2,... ,x_{2014}$ be real numbers different from $1$, such that $x_1 + x_2 +...+x_{2014} = 1$
and $\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+...+\frac{x_{2014}}{1-x_{2014}}=1$
What is the value of $\frac{x^2_1}{1-x_1}+\frac{x^2_2}{1-x_2}+...+\frac{x^2_{2014}}{1-x_{2014}}$ ?
2007 Nicolae Păun, 2
For a given natural number, $ n\ge 2, $ consider two matrices $ A,B\in\mathcal{M}_n(\mathbb{C}) $ that commute and such that $ A $ is invertible and that the function $ M:\mathbb{C}\longrightarrow\mathbb{C} ,M(x)=\det (A+xB) $ is bounded above or below.
Prove that $ B^n=0. $
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
OMMC POTM, 2024 7
Let $A$ and $B$ be two points on the same line $\ell$. If the points $P$ and $Q$ are two points $X$ on $\ell$ that mazimize and minimize the ratio $\frac{AX}{BX}$ respectively, prove that $A,B,P$ and $Q$ are concyclic.
2022 Purple Comet Problems, 15
Let $a$ be a real number such that $$5 \sin^4 \left( \frac{a}{2} \right)+ 12 \cos a = 5 cos^4 \left( \frac{a}{2} \right)+ 12 \sin a.$$ There are relatively prime positive integers $m$ and $n$ such that $\tan a = \frac{m}{n}$ . Find $10m + n$.
1996 Korea National Olympiad, 1
If you draw $4$ points on the unit circle, prove that you can always find two points where their distance between is less than $\sqrt{2}.$
2024 CCA Math Bonanza, L3.1
Byan rolls a $12$-sided die, a $14$-sided die, a $20$-sided die, and a $24$-sided die. The probability the sum of the numbers the die landed on is divisible by $7$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Lightning 3.1[/i]
2025 All-Russian Olympiad, 9.2
The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). The points of tangency of the circumcircles of triangles \(ABE\) and \(CDE\) with their common external tangents lie on a circle \(\omega\). The points of tangency of the circumcircles of triangles \(ADE\) and \(BCE\) with their common external tangents lie on a circle \(\gamma\). Prove that the centers of circles \(\omega\) and \(\gamma\) coincide.
2013 Junior Balkan Team Selection Tests - Romania, 1
If $a, b, c > 0$ satisfy $a + b + c = 3$, then prove that
$$\frac{a^2(b + 1)}{ ab + a + b} + \frac{b^2(c + 1)}{ bc + b + c} + \frac{c^2(a + 1)}{ ca + c + a} \ge 2$$
Mathematical Excalibur P322/Vol.14, no.2
2013 Stanford Mathematics Tournament, 1
Andrew flips a fair coin $5$ times, and counts the number of heads that appear. Beth flips a fair coin $6$ times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth.
2010 Poland - Second Round, 3
Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).
1992 IMO Shortlist, 2
Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation:
\[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\]
Prove that there exists a unique solution of this equation.
2018 Malaysia National Olympiad, A2
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?
2018 Junior Balkan Team Selection Tests - Romania, 1
Prove that a positive integer $A$ is a perfect square if and only if, for all positive integers $n$, at least one of the numbers $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.
2010 Denmark MO - Mohr Contest, 5
An equilateral triangle $ABC$ is given. With $BC$ as diameter, a semicircle is drawn outside the triangle. On the semicircle, points $D$ and $E$ are chosen such that the arc lengths $BD, DE$ and $EC$ are equal. Prove that the line segments $AD$ and $AE$ divide the side $BC$ into three equal parts.
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