Found problems: 85335
1996 IberoAmerican, 2
Three tokens $A$, $B$, $C$ are, each one in a vertex of an equilateral triangle of side $n$. Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case $n=3$
Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules:
[b](1) [/b]First $A$ moves, after that $B$ moves, and then $C$, by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other.
[b](2)[/b] Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there waiting its turn.
Show that for every positive integer $n$ it is possible to paint red all the sides of the little triangles.
2020 CHKMO, 1
Given that ${a_n}$ and ${b_n}$ are two sequences of integers defined by
\begin{align*}
a_1=1, a_2=10, a_{n+1}=2a_n+3a_{n-1} & ~~~\text{for }n=2,3,4,\ldots, \\
b_1=1, b_2=8, b_{n+1}=3b_n+4b_{n-1} & ~~~\text{for }n=2,3,4,\ldots.
\end{align*}
Prove that, besides the number $1$, no two numbers in the sequences are identical.
2016 Belarus Team Selection Test, 1
Find all functions $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ such that
$$f(x-2f(y))= xf(y)-yf(x)+g(x)$$ for all real $x,y$
2004 Purple Comet Problems, 9
How many positive integers less that $200$ are relatively prime to either $15$ or $24$?
2005 Gheorghe Vranceanu, 3
Prove by the method of induction that:
[b]a)[/b] $ a!b! $ divides $ (a+b)! , $ for any natural numbers $ a,b. $
[b]b)[/b] $ p $ divides $ (-1)^{k+1} +\binom{p-1}{k} , $ for any odd primes $ p $ and $ k\in\{ 0,1,\ldots ,p-1\} . $
2023 Belarusian National Olympiad, 8.5
In every cell of the table $3 \times 3$ a monomial with a positive coefficient is written (cells (1,1); (2,3); (3,2) have the degree of two, cells (1,2);(2,1);(3,3) have a degree of one, cells (3,1);(2,2);(1,3) have a constant).
Vuga added up monomials in every row and got three quadratic polynomials. It turned out that exactly $N$ of them have real roots. Leka added up monomials in every column and got three quadratic polynomials. It turned out that exactly $M$ of them have real roots.
Find the maximum possible value of $N-M$.
2015 Junior Balkan MO, 3
Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]
2006 Denmark MO - Mohr Contest, 4
Of the numbers $1, 2,3,..,2006$, ten different ones must be selected. Show that you can pick ten different numbers with a sum greater than $10039$ in more ways than you can select ten different numbers with a sum less than $10030$.
2019 USMCA, 5
The number $2019$ is written on a blackboard. Every minute, if the number $a$ is written on the board, Evan erases it and replaces it with a number chosen from the set
$$ \left\{ 0, 1, 2, \ldots, \left\lceil 2.01 a \right\rceil \right\} $$
uniformly at random. Is there an integer $N$ such that the board reads $0$ after $N$ steps with at least $99\%$ probability?
1983 AIME Problems, 14
In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.
[asy]unitsize(2.5mm);
defaultpen(linewidth(.8pt)+fontsize(12pt));
dotfactor=3;
pair O1=(0,0), O2=(12,0);
path C1=Circle(O1,8), C2=Circle(O2,6);
pair P=intersectionpoints(C1,C2)[0];
path C3=Circle(P,sqrt(130));
pair Q=intersectionpoints(C3,C1)[0];
pair R=intersectionpoints(C3,C2)[1];
draw(C1);
draw(C2);
//draw(O2--O1);
//dot(O1);
//dot(O2);
draw(Q--R);
label("$Q$",Q,N);
label("$P$",P,dir(80));
label("$R$",R,E);
//label("12",waypoint(O1--O2,0.4),S);[/asy]
1993 Hungary-Israel Binational, 5
In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.
Let $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?
2020 LMT Fall, A28 B30
Arthur has a regular 11-gon. He labels the vertices with the letters in $CORONAVIRUS$ in consecutive order. Every non-ordered set of 3 letters that forms an isosceles triangle is a member of a set $S$, i.e. $\{C, O, R\}$ is in $S$. How many elements are in $S$?
[i]Proposed by Sammy Chareny[/i]
1999 Vietnam National Olympiad, 3
Let $ S \equal{} \{0,1,2,\ldots,1999\}$ and $ T \equal{} \{0,1,2,\ldots \}.$ Find all functions $ f: T \mapsto S$ such that
[b](i)[/b] $ f(s) \equal{} s \quad \forall s \in S.$
[b](ii)[/b] $ f(m\plus{}n) \equal{} f(f(m)\plus{}f(n)) \quad \forall m,n \in T.$
2011 Indonesia MO, 6
Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.
2016 Purple Comet Problems, 3
Find the positive integer $n$ such that $10^n$ cubic centimeters is the same as 1 cubic kilometer.
2019 USA TSTST, 4
Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$, then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is at most $C$.
[i]Merlijn Staps[/i]
2011 Baltic Way, 9
Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:
[list]
[*]All squares touching the border of the grid are coloured black.
[*]No four squares forming a $2\times 2$ square are coloured in the same colour.
[*]No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching
squares have the same colour.[/list]
Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?
2005 Olympic Revenge, 1
Let $S=\{1,2,3,\ldots,n\}$, $n$ an odd number. Find the parity of number of permutations $\sigma : S \Rightarrow S$ such that the sequence defined by \[a(i)=|\sigma(i)-i|\] is monotonous.
2017 Moldova Team Selection Test, 11
Find all ordered pairs of nonnegative integers $(x,y)$ such that
\[x^4-x^2y^2+y^4+2x^3y-2xy^3=1.\]
2022 MMATHS, 11
Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let $M$ and $N$ be the expected number of times Josh and Ron tap their screens, respectively. What is $|M-N|$?
PEN N Problems, 7
Let $\{n_{k}\}_{k \ge 1}$ be a sequence of natural numbers such that for $i<j$, the decimal representation of $n_{i}$ does not occur as the leftmost digits of the decimal representation of $n_{j}$. Prove that \[\sum^{\infty}_{k=1}\frac{1}{n_{k}}\le \frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{9}.\]
1994 India Regional Mathematical Olympiad, 5
Let $A$ be a set of $16$ positive integers with the property that the product of any two distinct members of $A$ will not exceed 1994. Show that there are numbers $a$ and $b$ in the set $A$ such that the gcd of $a$ and $b$ is greater than 1.
2013 Spain Mathematical Olympiad, 5
Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions
$i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct).
$ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$
2023/2024 Tournament of Towns, 5
Chord $D E$ of the circumcircle of the triangle $A B C$ intersects sides $A B$ and $B C$ in points $P$ and $Q$ respectively. Point $P$ lies between $D$ and $Q$. Angle bisectors $D F$ and $E G$ are drawn in triangles $A D P$ and $Q E C$. It turned out that the points $D$, $F, G, E$ are concyclic. Prove that the points $A, P, Q, C$ are concyclic.
Azamat Mardanov
2019 Estonia Team Selection Test, 6
It is allowed to perform the following transformations in the plane with any integers $a$:
(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,
(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.
Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:
a) Vertices of a square,
b) Vertices of a rectangle with unequal side lengths?