Found problems: 85335
1984 Iran MO (2nd round), 2
Consider the function
\[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\]
Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.
2006 Paraguay Mathematical Olympiad, 1
What are the last two digits of the decimal representation of $21^{2006}$?
2016 Federal Competition For Advanced Students, P1, 2
We are given an acute triangle $ABC$ with $AB > AC$ and orthocenter $H$. The point $E$ lies symmetric to $C$ with respect to the altitude $AH$. Let $F$ be the intersection of the lines $EH$ and $AC$. Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$.
(Karl Czakler)
2019 Middle European Mathematical Olympiad, 2
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$.
[i]Proposed by Walther Janous, Austria[/i]
1997 Belarusian National Olympiad, 4
A set $M$ consists of $n$ elements. Find the greatest $k$ for which there is a collection of $k$ subsets of $M$ such that for any subsets $A_{1},...,A_{j}$ from the collection, there is an element belonging to an odd number of them
2020 Romanian Master of Mathematics Shortlist, C3
Determine the smallest positive integer $k{}$ satisfying the following condition: For any configuration of chess queens on a $100 \times 100$ chequered board, the queens can be coloured one of $k$ colours so that no two queens of the same colour attack each other.
[i]Russia, Sergei Avgustinovich and Dmitry Khramtsov[/i]
2015 Costa Rica - Final Round, G4
Consider $\vartriangle ABC$, right at $B$, let $I$ be its incenter and $F,D,E$ the points where the circle inscribed on sides AB, $BC$ and $AC$, respectively. If $M$ is the intersection point of $CI$ and $EF$, and $N$ is the intersection point of $DM$ and $AB$. Prove that $AN = ID$.
2008 IMO, 1
Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$.
Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.
[i]Author: Andrey Gavrilyuk, Russia[/i]
2017 Purple Comet Problems, 29
Find the number of three-element subsets of $\{1, 2, 3,...,13\}$ that contain at least one element that is a multiple of $2$, at least one element that is a multiple of $3$, and at least one element that is a multiple of $5$ such as $\{2,3, 5\}$ or $\{6, 10,13\}$.
2008 Germany Team Selection Test, 1
Show that there is a digit unequal to 2 in the decimal represesentation of $ \sqrt [3]{3}$ between the $ 1000000$-th und $ 3141592$-th position after decimal point.
2023 Bulgarian Spring Mathematical Competition, 10.2
An isosceles $\triangle ABC$ has $\angle BAC =\angle ABC =72^{o}$. The angle bisector $AL$ meets the line through $C$ parallel to $AB$ at $D$.
$a)$ Prove that the circumcenter of $\triangle ADC$ lies on $BD$.
$b)$ Prove that $\frac {BE} {BL}$ is irrational.
2021 Thailand TSTST, 1
Let $a,b,c$ be distinct positive real numbers such that $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\leq 1$. Prove that $$2\left(\sqrt{\frac{a+b}{ac}}+\sqrt{\frac{b+c}{ba}}+\sqrt{\frac{c+a}{cb}}\right)<\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)}.$$
2010 IMO Shortlist, 2
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.
[i]Proposed by Tonći Kokan, Croatia[/i]
1993 All-Russian Olympiad Regional Round, 9.8
Number $ 0$ is written on the board. Two players alternate writing signs and numbers to the right, where the first player always writes either $ \plus{}$ or $ \minus{}$ sign, while the second player writes one of the numbers $ 1, 2, ... , 1993$,writing each of these numbers exactly once. The game ends after $ 1993$ moves. Then the second player wins the score equal to the absolute value of the expression obtained thereby on the board. What largest score can he always win?
2007 Princeton University Math Competition, 3
Is there a set of distinct integers $X$ containing all the primes less than $2007$ such that the product of the elements of $X$ equals the sum of the squares of those elements?
2007 Nicolae Păun, 1
Let be nine nonzero decimal digits $ a_1,a_2,a_3,b_1,b_2,b_3,c_1,c_2,c_3 $ chosen such that the polynom
$$ \left( 100a_1+10a_2+a_3 \right) X^2 +\left( 100b_1+10b_2+b_3 \right) X +100c_1+10c_2+c_3 $$
admits at least a real solution.
Prove that at least one of the polynoms $ a_iX^2+b_iX+c_i\quad (i\in\{1,2,3\}) $ admits at least a real solution.
[i]Nicolae Mușuroia[/i]
2019 ELMO Shortlist, C5
Given a permutation of $1,2,3,\dots,n$, with consecutive elements $a,b,c$ (in that order), we may perform either of the [i]moves[/i]:
[list]
[*] If $a$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $b,c,a$ (in that order)
[*] If $c$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $c,a,b$ (in that order)
[/list]
What is the least number of sets in a partition of all $n!$ permutations, such that any two permutations in the same set are obtainable from each other by a sequence of moves?
[i]Proposed by Milan Haiman[/i]
2022 Kosovo & Albania Mathematical Olympiad, 4
Let $A$ be the set of natural numbers $n$ such that the distance of the real number $n\sqrt{2022} - \frac13$ from the nearest integer is at most $\frac1{2022}$. Show that the equation $$20x + 21y = 22z$$ has no solutions over the set $A$.
2008 Kazakhstan National Olympiad, 1
Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$?
Remark: Two cells are called connected if they have a common edge.
2007 Gheorghe Vranceanu, 3
Let be a function $ s:\mathbb{N}^2\longrightarrow \mathbb{N} $ that sends $ (m,n) $ to the number of solutions in $ \mathbb{N}^n $ of the equation:
$$ x_1+x_2+\cdots +x_n=m $$
[b]1)[/b] Prove that:
$$ s(m+1,n+1)=s(m,n)+s(m,n+1) =\prod_{r=1}^n\frac{m-r+1}{r} ,\quad\forall m,n\in\mathbb{N} $$
[b]2)[/b] Find $ \max\left\{ a_1a_2\cdots a_{20}\bigg| a_1+a_2+\cdots +a_{20}=2007, a_1,a_2,\ldots a_{20}\in\mathbb{N} \right\} . $
2010 HMNT, 8
Allison has a coin which comes up heads $\frac23$ of the time. She flips it $5$ times. What is the probability that she sees more heads than tails?
2018 Balkan MO, 4
Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$
Proposed by Stanislav Dimitrov,Bulgaria
2014 Irish Math Olympiad, 3
In the triangle ABC, D is the foot of the altitude from A to BC, and M is the midpoint of the line
segment BC. The three angles ∠BAD, ∠DAM and ∠MAC are all equal. Find the angles of the
triangle ABC.
2021 OMMock - Mexico National Olympiad Mock Exam, 2
For which positive integers $n$ does there exist a positive integer $m$ such that among the numbers $m + n, 2m + (n - 1), \dots, nm + 1$, there are no two that share a common factor greater than $1$?
1972 Canada National Olympiad, 9
Four distinct lines $L_1,L_2,L_3,L_4$ are given in the plane: $L_1$ and $L_2$ are respectively parallel to $L_3$ and $L_4$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.