Found problems: 85335
2018 China Team Selection Test, 5
Let $ABC$ be a triangle with $\angle BAC > 90 ^{\circ}$, and let $O$ be its circumcenter and $\omega$ be its circumcircle. The tangent line of $\omega$ at $A$ intersects the tangent line of $\omega$ at $B$ and $C$ respectively at point $P$ and $Q$. Let $D,E$ be the feet of the altitudes from $P,Q$ onto $BC$, respectively. $F,G$ are two points on $\overline{PQ}$ different from $A$, so that $A,F,B,E$ and $A,G,C,D$ are both concyclic. Let M be the midpoint of $\overline{DE}$. Prove that $DF,OM,EG$ are concurrent.
2000 Moldova National Olympiad, Problem 8
Two circles intersect at $M$ and $N$. A line through $M$ meets the circles at $A$ and $B$, with $M$ between $A$ and $B$. Let $C$ and $D$ be the midpoints of the arcs $AN$ and $BN$ not containing $M$, respectively, and $K$ and $L$ be the midpoints of $AB$ and $CD$, respectively. Prove that $CL=KL$.
1968 IMO Shortlist, 6
If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation
\[\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n\]
has at least $n - 1$ real roots.
2023 Math Prize for Girls Problems, 16
Let $f(x) = x^2 - 3/4$. Let $f^{(n)}(x)$ denote the composition of $f$ with itself $n$ times. For example, $f^{(3)}(x) = f(f(f(x)))$. Let $R$ be the set of complex numbers that is the union of the roots of the polynomials
$f^{(n)}(x^2 + 3/4)$ over positive integers $n$. Let $B$ be the smallest rectangle in the complex plane with sides parallel to the real and imaginary axes that contains $R$. What is the square of the area of $B$?
2018 Thailand Mathematical Olympiad, 6
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ .
a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$.
b) Show that $A$ is an infinite set.
2008 May Olympiad, 3
On a blackboard are written all the integers from $1$ to $2008$ inclusive. Two numbers are deleted and their difference is written. For example, if you erase $5$ and $241$, you write $236$. This continues, erasing two numbers and writing their difference, until only one number remains. Determine if the number left at the end can be $2008$. What about $2007$? In each case, if the answer is affirmative, indicate a sequence with that final number, and if it is negative, explain why.
2019 Purple Comet Problems, 14
For real numbers $a$ and $b$, let $f(x) = ax + b$ and $g(x) = x^2 - x$. Suppose that $g(f(2)) = 2, g(f(3)) = 0$, and $g(f(4)) = 6$. Find $g(f(5))$.
2012 Serbia Team Selection Test, 3
Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$.
a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic.
b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.
2012 Iran MO (3rd Round), 6
[b]a)[/b] Prove that $a>0$ exists such that for each natural number $n$, there exists a convex $n$-gon $P$ in plane with lattice points as vertices such that the area of $P$ is less than $an^3$.
[b]b)[/b] Prove that there exists $b>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $bn^2$.
[b]c)[/b] Prove that there exist $\alpha,c>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $cn^{2+\alpha}$.
[i]Proposed by Mostafa Eynollahzade[/i]
2016 Indonesia TST, 3
Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, $i = 1, 2, \dots, m$. It is known that every two elements of $X$ is contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.
2022 Bosnia and Herzegovina IMO TST, 4
In each square of a $4 \times 4$ table a number $0$ or $1$ is written, such that the product of every two neighboring squares is $0$ (neighboring by side).
$a)$ In how many ways is this possible to do if the middle $2\times 2$ is filled with $4$ zeros?
$b)$ In general, in how many ways is this possible to do (regardless of the middle $2 \times 2$)?
2014 Germany Team Selection Test, 3
Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$.
Prove that if $2013$ is [i]good[/i], then so is $20$.
2002 Junior Balkan Team Selection Tests - Moldova, 8
Find all triplets (a, b, c) of positive integers so that
$a^2b$, $b^2c$ and $c^2a$ devide $a^3+b^3+c^3$
2022 Macedonian Team Selection Test, Problem 3
We consider all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(f(n)+n)=n$ and $f(a+b-1) \leq f(a)+f(b)$ for all positive integers $a, b, n$. Prove that there are at most two values for $f(2022)$.
$\textit {Proposed by Ilija Jovcheski}$
2004 AMC 10, 22
A triangle with sides of $ 5$, $ 12$, and $ 13$ has both an inscibed and a circumscribed circle. What is the distance between the centers of those circles?
$ \textbf{(A)}\ \frac{3\sqrt{5}}{2}\qquad
\textbf{(B)}\ \frac{7}{2}\qquad
\textbf{(C)}\ \sqrt{15}\qquad
\textbf{(D)}\ \frac{\sqrt{65}}{2}\qquad
\textbf{(E)}\ \frac{9}{2}$
2007 Grigore Moisil Intercounty, 3
[b]a)[/b] Let $ AA',BB',CC' $ be the altitudes of a triangle $ ABC. $ Prove that
$$ \frac{BC}{AA'}\cdot \overrightarrow{AA'} +\frac{AC}{BB'}\cdot \overrightarrow{BB'} +\frac{AB}{CC'}\cdot \overrightarrow{CC'} =0. $$
[b]b)[/b] The sum of the vectors that are perpendicular to the sides of a convex polygon and have equal lengths as those sides, respectively, is $ 0. $
2004 Regional Olympiad - Republic of Srpska, 3
Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite.
If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?
2009 Postal Coaching, 3
Let $ABC$ be a triangle with circumcentre $O$ and incentre $I$ such that $O$ is different from $I$. Let $AK, BL, CM$ be the altitudes of $ABC$, let $U, V , W$ be the mid-points of $AK, BL, CM$ respectively. Let $D, E, F$ be the points at which the in-circle of $ABC$ respectively touches the sides $BC, CA, AB$. Prove that the lines $UD, VE, WF$ and $OI$ are concurrent.
2012 Junior Balkan Team Selection Tests - Romania, 1
Prove that if the positive real numbers $p$ and $q$ satisfy $\frac{1}{p}+\frac{1}{q}= 1$, then
a) $\frac{1}{3} \le \frac{1}{p (p + 1)} +\frac{1}{q (q + 1)} <\frac{1}{2}$
b) $\frac{1}{p (p - 1)} + \frac{1}{q (q - 1)} \ge 1$
1997 Tournament Of Towns, (533) 5
Prove that the number
(a) $97^{97}$
(b) $1997^{17}$
cannot be equal to a sum of cubes of several consecutive integers.
(AA Egorov)
2017 Estonia Team Selection Test, 6
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$
2021 DIME, 10
There exist complex numbers $z_1,z_2,\dots,z_{10}$ which satisfy$$|z_ki^k+ z_{k+1}i^{k+1}| = |z_{k+1}i^k+ z_ki^{k+1}|$$for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. If $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1|+|z_2|+\cdots+|z_{10}|$.
[i]Proposed by DeToasty3[/i]
2010 Tournament Of Towns, 1
Is it possible to split all straight lines in a plane into the pairs of perpendicular lines, so that every line belongs to a single pair?
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 $q$.
2017 Online Math Open Problems, 23
Call a nonempty set $V$ of nonzero integers \emph{victorious} if there exists a polynomial $P(x)$ with integer coefficients such that $P(0)=330$ and that $P(v)=2|v|$ holds for all elements $v\in V$. Find the number of victorious sets.
[i]Proposed by Yannick Yao[/i]