Found problems: 85335
2019 Saudi Arabia Pre-TST + Training Tests, 1.2
Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .
2015 Princeton University Math Competition, A5/B7
Alice has an orange $\text{3-by-3-by-3}$ cube, which is comprised of $27$ distinguishable, $\text{1-by-1-by-1}$ cubes. Each small cube was initially orange, but Alice painted $10$ of the small cubes completely black. In how many ways could she have chosen $10$ of these smaller cubes to paint black such that every one of the $27$ $\text{3-by-1-by-1}$ sub-blocks of the $\text{3-by-3-by-3}$ cube contains at least one small black cube?
2012 Turkey MO (2nd round), 3
Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.
LMT Accuracy Rounds, 2021 F7
Find the number of ways to tile a $12 \times 3$ board with $1 \times 4$ and $2 \times 2$ tiles with no overlap or uncovered space.
2009 Belarus Team Selection Test, 2
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
2008 JBMO Shortlist, 6
If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $1 + ab + bc + cd + da + ac + bd > a + b + c + d$.
1996 Taiwan National Olympiad, 5
Dertemine integers $a_{1},a_{2},...,a_{99}=a_{0}$ satisfying $|a_{k}-a_{k-1}|\geq 1996$ for all $k=1,2,...,99$, such that $m=\max_{1\leq k\leq 99} |a_{k}-a_{k-1}|$ is minimum possible, and find the minimum value $m^{*}$ of $m$.
2010 Contests, 3
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be externally tangent at a point $A$. A line tangent to $\mathcal{C}_1$ at $B$ intersects $\mathcal{C}_2$ at $C$ and $D$; then the segment $AB$ is extended to intersect $\mathcal{C}_2$ at a point $E$. Let $F$ be the midpoint of $\overarc{CD}$ that does not contain $E$, and let $H$ be the intersection of $BF$ with $\mathcal{C}_2$. Show that $CD$, $AF$, and $EH$ are concurrent.
ICMC 4, 2
Let $p > 3$ be a prime number. A sequence of $p-1$ integers $a_1,a_2, \dots, a_{p-1}$ is called [i]wonky[/i] if they are distinct modulo \(p\) and $a_ia_{i+2} \not\equiv a_{i+1}^2 \pmod p$ for all \(i \in \{1, 2, \dots, p-1\}\), where \(a_p = a_1\) and \(a_{p+1} = a_2\). Does there always exist a wonky sequence such that $$a_1a_2, \qquad a_1a_2+a_2a_3, \qquad \dots, \qquad a_1a_2+\cdots +a_{p-1}a_1,$$ are all distinct modulo $p$?
[i]Proposed by Harun Khan[/i]
2014 Contests, 3
Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$.
Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other.
Note: the pieces can be rotated and flipped over.
2016 Balkan MO Shortlist, C3
The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of $1201$ colours so that no rectangle with perimeter $100$ contains two squares of the same colour. Show that no rectangle of size $1\times1201$ or $1201\times1$ contains two squares of the same colour.
[i]Note: Any rectangle is assumed here to have sides contained in the lines of the grid.[/i]
[i](Bulgaria - Nikolay Beluhov)[/i]
1990 Tournament Of Towns, (256) 4
A set of $103$ coins that look alike is given. Two coins (whose weights are equal) are counterfeit. The other $101$ (genuine) coins also have the same weight, but a different weight from that of the counterfeit coins. However it is not known whether it is the genuine coins or the counterfeit coins which are heavier. How can this question be resolved by three weighings on the one balance? (It is not required to separate the counterfeit coins from the genuine ones.)
(D. Fomin, Leningrad)
2011 Irish Math Olympiad, 1
Suppose $abc\neq 0$. Express in terms of $a,b,$ and $c$ the solutions $x,y,z,u,v,w$ of the equations $$x+y=a,\quad z+u=b,\quad v+w=c,\quad ay=bz,\quad ub=cv,\quad wc=ax.\quad$$
2005 Moldova National Olympiad, 10.7
Determine all strictly increasing functions $ f: R\rightarrow R$ satisfying relationship $ f(x\plus{}f(y))\equal{}f(x\plus{}y)\plus{}2005$
for any real values of x and y.
1998 AMC 8, 22
Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.
Rule 1: If the integer is less than 10, multiply it by 9.
Rule 2: If the integer is even and greater than 9, divide it by 2.
Rule 3: If the integer is odd and greater than 9, subtract 5 from it.
Find the $98th$ term of the sequence that begins $ 98, 49,\ldots . $
$ \text{(A)}\ 6\qquad\text{(B)}\ 11\qquad\text{(C)}\ 22\qquad\text{(D)}\ 27\qquad\text{(E)}\ 54 $
2007 Hanoi Open Mathematics Competitions, 6
In triangle $ABC, \angle BAC = 60^o, \angle ACB = 90^o$ and $D$ is on $BC$.
If $AD$ bisects $\angle BAC$ and $CD = 3$ cm, calculate $DB$ .
2006 India Regional Mathematical Olympiad, 1
Let $ ABC$ be an acute-angled triangle and let $ D,E,F$ be the feet of perpendiculars from $ A,B,C$ respectively to $ BC,CA,AB .$ Let the perpendiculars from $ F$ to $ CB,CA,AD,BE$ meet them in $ P,Q,M,N$ respectively. Prove that the points $ P,Q,M,N$ are collinear.
2009 Junior Balkan Team Selection Test, 2
From the set $ \{1,2,3,\ldots,2009\}$ we choose $ 1005$ numbers, such that sum of any $ 2$ numbers isn't neither $ 2009$ nor $ 2010$. Find all ways on we can choose these $ 1005$ numbers.
1985 AIME Problems, 11
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?
2019 Brazil National Olympiad, 3
Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the
vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre
2009 All-Russian Olympiad Regional Round, 9.8
8 chess players participated in the chess tournament and everyone played exactly one game with everyone else. It is known that any two chess players who play a draw with each other ended up scoring different numbers of points. Find the greatest possible number of draws in this tournament. (For winning the game the chess player is awarded $1$ point, for a draw $1/2$ points, for defeat $0$ points.)
2018 PUMaC Live Round, Estimation 2
How many perfect squares have the digits $1$ through $9$ each exactly once when written in base $10$?
You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, your score will be $\lfloor12.5\cdot\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor.$
1970 Poland - Second Round, 4
Prove that if triangle $T_1$ contains triangle $T_2$, then the perimeter of triangle $T_1$ is not less than the perimeter of triangle $T_2$.
2010 Miklós Schweitzer, 1
Let $ p $ be prime. Denote by $ N (p) $ the number of integers $ x $ for which $ 1 \leq x \leq p $ and
$$
x ^ {x} \equiv 1 \quad (\bmod p)
$$Prove that there exist numbers $ c <1/2 $ and $ p_ {0}> 0 $ such that
$$
N (p) \leq p ^ {c}
$$if $ p \ge p_ {0} $.
2022 Serbia National Math Olympiad, P2
Let $a$, $b$ and $c$ be positive real numbers and $a^3+b^3+c^3=3$. Prove
$$\frac{1}{3-2a}+\frac{1}{3-2b}+\frac{1}{3-2c}\geq 3$$