Found problems: 85335
2014 Harvard-MIT Mathematics Tournament, 8
Let $ABC$ be an acute triangle with circumcenter $O$ such that $AB=4$, $AC=5$, and $BC=6$. Let $D$ be the foot of the altitude from $A$ to $BC$, and $E$ be the intersection of $AO$ with $BC$. Suppose that $X$ is on $BC$ between $D$ and $E$ such that there is a point $Y$ on $AD$ satisfying $XY\parallel AO$ and $YO\perp AX$. Determine the length of $BX$.
2017 IMEO, 3
A triple $(x,y,z)$ of real numbers is called a [i]superparticular[/i] if
$$\frac{x+1}{x} \cdot \frac{y+1}{y}=\frac{z+1}{z}$$
Find all superparticular positive integer triples.
2018 Saint Petersburg Mathematical Olympiad, 6
$a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$.
1968 Putnam, A4
Let $S^{2}\subset \mathbb{R}^{3}$ be the unit sphere. Show that for any $n$ points on $ S^{2}$, the sum of the squares of the $\frac{n(n-1)}{2}$ distances between them is at most $n^{2}$.
2016 NZMOC Camp Selection Problems, 4
A quadruple $(p, a, b, c)$ of positive integers is a[i] karaka quadruple[/i] if
$\bullet$ $p$ is an odd prime number
$\bullet$ $a, b$ and $c$ are distinct, and
$\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$.
(a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$.
(b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$
2022 JHMT HS, 3
Triangle $WSE$ has side lengths $WS=13$, $SE=15$, and $WE=14$. Points $J$ and $H$ lie on $\overline{SE}$ such that $SJ=JH=HE=5$. Let the angle bisector of $\angle{WES}$ intersect $\overline{WH}$ and $\overline{WJ}$ at points $M$ and $T$, respectively. Find the area of quadrilateral $JHMT$.
2018 BAMO, B
A square with sides of length $1$ cm is given. There are many different ways to cut the square into four rectangles.
Let $S$ be the sum of the four rectangles’ perimeters. Describe all possible values of $S$ with justification.
2019 Korea - Final Round, 3
Prove that there exist infinitely many positive integers $k$ such that the sequence $\{x_n\}$ satisfying
$$ x_1=1, x_2=k+2, x_{n+2}-(k+1)x_{n+1}+x_n=0(n \ge 0)$$
does not contain any prime number.
2014 Saudi Arabia Pre-TST, 4.4
Let $\vartriangle ABC$ be an acute triangle, with $\angle A> \angle B \ge \angle C$. Let $D, E$ and $F$ be the tangency points between the incircle of triangle and sides $BC, CA, AB$, respectively. Let $J$ be a point on $(BD)$, $K$ a point on $(DC)$, $L$ a point on $(EC)$ and $M$ a point on $(FB)$, such that $$AF = FM = JD = DK = LE = EA.$$Let $P$ be the intersection point between $AJ$ and $KM$ and let $Q$ be the intersection point between $AK$ and $JL$. Prove that $PJKQ$ is cyclic.
2018 AMC 12/AHSME, 25
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?
$\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$
2004 Germany Team Selection Test, 3
Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$.
Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i]
[i]Proposed by Laurentiu Panaitopol, Romania[/i]
2018 China Team Selection Test, 2
Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.
2018 Junior Balkan Team Selection Tests - Romania, 3
Let $ABCD$ be a cyclic quadrilateral. The line parallel to $BD$ passing through $A$ meets the line parallel to $AC$ passing through $B$ at $E$. The circumcircle of triangle $ABE$ meets the lines $EC$ and $ED$, again, at $F$ and $G$, respectively. Prove that the lines $AB, CD$ and $FG$ are either parallel or concurrent.
2023 Bangladesh Mathematical Olympiad, P6
Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.
2005 Kyiv Mathematical Festival, 1
On Monday a school library was attended by 5 students, on Tuesday, by 6, on Wednesday, by 4, on Thursday, by 8, and on Friday, by 7. None of them have attended the library two days running. What is the least possible number of students who visited the library during a week?
1997 Miklós Schweitzer, 10
Assign independent standard normally distributed random variables to the vertices of an n-dimensional cube. Say one vertex is greater than another if the assigned number is greater. Define a random walk on the vertices according to the following rules:
a) the starting point is chosen from all the vertices with equal probability,
b) during our journey, if we reach a vertex such that there are adjacent vertices which have higher values, we choose the next vertex with equal probability,
c) if there is none, we stop.
Prove that $\forall\varepsilon>0 \,\exists K\, \forall n>1$
$$P(\lambda> K \log n) <\varepsilon$$
where $\lambda$ is the number of steps of the random walk.
1968 All Soviet Union Mathematical Olympiad, 114
Given a quadrangle $ABCD$. The lengths of all its sides and diagonals are the rational numbers. Let $O$ be the point of its diagonals intersection. Prove that $|AO|$ - the length of the $[AO]$ segment is also rational.
2000 239 Open Mathematical Olympiad, 3
Let $ AA_1 $ and $ CC_1 $ be the altitudes of the acute-angled triangle $ ABC $. A line passing through the centers of the inscribed circles the triangles $ AA_1C $ and $ CC_1A $ intersect the sides of $ AB $ and $ BC $ triangle $ ABC $ at points $ X $ and $ Y $. Prove that $ BX = BY $.
2019 Peru IMO TST, 6
Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that:
[LIST]
[*] The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one. [/*]
[*] The sets $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one. [/*]
[*] For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$.
[/LIST]
2020 Bulgaria Team Selection Test, 3
Let $\mathcal{C}$ be a family of subsets of $A=\{1,2,\dots,100\}$ satisfying the following two conditions:
1) Every $99$ element subset of $A$ is in $\mathcal{C}.$
2) For any non empty subset $C\in\mathcal{C}$ there is $c\in C$ such that $C\setminus\{c\}\in \mathcal{C}.$
What is the least possible value of $|\mathcal{C}|$?
2023 Czech-Polish-Slovak Junior Match, 3
$n$ people met at the party, with $n \ge 2$. Each person dislikes exactly one other person present at the party (but not necessarily reciprocal, i.e. it may happen that $A$ dislikes $B$ even though $B$ does not dislike $A$) and likes all others. Prove that guests can be seated at three tables in such a way that each guest likes all the people at his table.
2013 South africa National Olympiad, 2
A is a two-digit number and B is a three-digit number such that A increased by B% equals B reduced by A%. Find all possible pairs (A, B).
Denmark (Mohr) - geometry, 1992.2
In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.
2014 Contests, 3
Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$
2023 Malaysian Squad Selection Test, 1
Ivan has a $m \times n$ board, and he color some squares black, so that no three black squares form a L-triomino up to rotations and reflections. What is the maximal number of black squares that Ivan can color?
[i]Proposed by Ivan Chan Kai Chin[/i]