Found problems: 85335
2022 Bulgaria National Olympiad, 4
Let $n\geq 4$ be a positive integer and $x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2}$ be real numbers such that $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$. If there exists an $a>0$ such that
\[x_{i}^2=a+x_{i+1}x_{i+2}\quad\forall 1\leq i\leq n\]
then prove that at least $2$ of the numbers $x_{1},x_{2},\ldots ,x_{n}$ are negative.
2015 Korea - Final Round, 5
For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$.
They are defined inductively, by the following recurrences.
$A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$
$B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$
Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.
2014 Online Math Open Problems, 27
Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$.
Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying
\[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \]
for all $x,y \in S$.
Let $N$ be the product of all possible nonzero values of $f(81)$.
Find the remainder when when $N$ is divided by $p$.
[i]Proposed by Yang Liu and Ryan Alweiss[/i]
2010 Miklós Schweitzer, 5
Given the vectors $ v_ {1}, \dots, v_ {n} $ and $ w_ {1}, \dots, w_ {n} $ in the plane with the following properties:
for every $ 1 \leq i \leq n $ ,$ \left | v_{i} -w_{i} \right | \leq 1, $ and for every $ 1 \leq i <j \leq n $ ,$ \left | v_{i} -v_{j} \right | \ge 3 $ and $ v_{i} -w_ {i} \ne v_ {j} -w_ {j} $. Prove that for sets $ V = \left \{v_ {1}, \dots, v_{n } \right \} $ and $ W = \left \{w_ {1}, \dots, w_ {n} \right \}$, the set of $ V + (V \cup W) $ must have at least $ cn^{3/2} $ elements ,for some universal constant $ c>0 $ .
1976 IMO Longlists, 23
Prove that in a Euclidean plane there are infinitely many concentric circles $C$ such that all triangles inscribed in $C$ have at least one irrational side.
Ukraine Correspondence MO - geometry, 2003.5
Let $O$ be the center of the circle $\omega$, and let $A$ be a point inside this circle, different from $O$. Find all points $P$ on the circle $\omega$ for which the angle $\angle OPA$ acquires the greatest value.
2013 VJIMC, Problem 2
An $n$-dimensional cube is given. Consider all the segments connecting any two different vertices of the cube. How many distinct intersection points do these segments have (excluding the vertices)?
2017 India Regional Mathematical Olympiad, 5
Let \(\Omega\) be a circle with a chord \(AB\) which is not a diameter. \(\Gamma_{1}\) be a circle on one side of \(AB\) such that it is tangent to \(AB\) at \(C\) and internally tangent to \(\Omega\) at \(D\). Likewise, let \(\Gamma_{2}\) be a circle on the other side of \(AB\) such that it is tangent to \(AB\) at \(E\) and internally tangent to \(\Omega\) at \(F\). Suppose the line \(DC\) intersects \(\Omega\) at \(X \neq D\) and the line \(FE\) intersects \(\Omega\) at \(Y \neq F\). Prove that \(XY\) is a diameter of \(\Omega\) .
2022 Saudi Arabia JBMO TST, 2
Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$
1976 Bundeswettbewerb Mathematik, 2
Two congruent squares $Q$ and $Q'$ are given in the plane. Show that they can be divided into parts $T_1, T_2, \ldots , T_n$ and $T'_1 , T'_2 , \ldots , T'_n$, respectively, such that $T'_i$ is the image of $T_i$ under a translation for $i=1,2, \ldots, n.$
1991 Arnold's Trivium, 78
Solve the Cauchy problem
\[\frac{\partial ^2A}{\partial t^2}=9\frac{\partial^2 A}{\partial x^2}-2B,\;\frac{\partial^2 B}{\partial t^2}=6\frac{\partial^2 B}{\partial x^2}-2A\]
\[A|_{t=0}=\cos x,\; B|_{t=0}=0,\; \left.\frac{\partial A}{\partial t}\right|_{t=0}=\left.\frac{\partial B}{\partial t}\right|_{t=0}=0\]
2017 Kosovo National Mathematical Olympiad, 5
A sphere with ray $R$ is cut by two parallel planes. such that the center of the sphere is outside the region determined by these planes. Let $S_{1}$ and $S_{2}$ be the areas of the intersections, and $d$ the distance between these planes. Find the area of the intersection of the sphere with the plane parallel with these two planes, with equal distance from them.
1984 Poland - Second Round, 3
The given sequences are $ (x_1, x_2, \ldots, x_n) $, $ (y_1, y_2, \ldots, y_n) $ with positive terms. Prove that there exists a permutation $ p $ of the set $ \{1, 2, \ldots, n\} $ such that for every real $ t $ the sequence
$$ (x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })$$ has the following property: there is a number $ k $ such that $ 1 \leq k \leq n $ and all non-zero terms of the sequence with indices less than $ k $ are of the same sign and all non-zero terms of the sequence with indices not less than $ k $ are the same sign.
2014 Contests, 2
A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$.
Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.
1999 USAMTS Problems, 2
Let $N=111...1222...2$, where there are $1999$ digits of $1$ followed by $1999$ digits of $2$. Express $N$ as the product of four integers, each of them greater than $1$.
2010 All-Russian Olympiad Regional Round, 11.6
At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .
2012 Bogdan Stan, 4
Let $ D $ be a point on the side $ BC $ (excluding its endpoints) of a triangle $ ABC $ with $ AB>AC, $ such that $ \frac{\angle BAD}{\angle BAC} $ is a rational number. Prove the following:
$$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$
2007 JBMO Shortlist, 1
Find all the pairs positive integers $(x, y)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}$ ,
where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.
2011 NZMOC Camp Selection Problems, 5
Let a square $ABCD$ with sides of length $1$ be given. A point $X$ on $BC$ is at distance $d$ from $C$, and a point $Y$ on $CD$ is at distance $d$ from $C$. The extensions of: $AB$ and $DX$ meet at $P$, $AD$ and $BY$ meet at $Q, AX$ and $DC$ meet at $R$, and $AY$ and $BC$ meet at $S$. If points $P, Q, R$ and $S$ are collinear, determine $d$.
2007 Harvard-MIT Mathematics Tournament, 36
[i]The Marathon.[/i] Let $\omega$ denote the incircle of triangle $ABC$. The segments $BC$, $CA$, and $AB$ are tangent to $\omega$ at $D$, $E$ and $F$, respectively. Point $P$ lies on $EF$ such that segment $PD$ is perpendicular to $BC$. The line $AP$ intersects $BC$ at $Q$. The circles $\omega_1$ and $\omega_2$ pass through $B$ and $C$, respectively, and are tangent to $AQ$ at $Q$; the former meets $AB$ again at $X$, and the latter meets $AC$ again at $Y$. The line $XY$ intersects $BC$ at $Z$. Given that $AB=15$, $BC=14$, and $CA=13$, find $\lfloor XZ\cdot YZ\rfloor$.
2024 IFYM, Sozopol, 5
The function $f: A \rightarrow A$ is such that $f(x) \leq x^2 \mbox{ and } f(x+y) \leq f(x) + f(y) + 2xy$ for any $x, y \in A$.
a) If $A = \mathbb{R}$, find all functions satisfying the conditions.
b) If $A = \mathbb{R}^{-}$, prove that there are infinitely many functions satisfying the conditions.
[i](With $\mathbb{R}^{-}$ we denote the set of negative real numbers.)[/i]
2012 Pre-Preparation Course Examination, 5
Suppose that for the linear transformation $T:V \longrightarrow V$ where $V$ is a vector space, there is no trivial subspace $W\subset V$ such that $T(W)\subseteq W$. Prove that for every polynomial $p(x)$, the transformation $p(T)$ is invertible or zero.
2016 Denmark MO - Mohr Contest, 4
Alma and Bertha play the following game. There are $100$ round, $200$ triangular and $200$ square pieces on a table. In each move a player must remove two pieces, but it cannot be a triangle and a square. Alma starts, and one loses if one is unable to move or if there are no pieces left when it is one’s turn. Which player has a winning strategy?
2003 AMC 8, 3
A burger at Ricky C's weighs 120 grams, of which 30 grams are filler. What percent of the burger is not filler?
$\textbf{(A)}\ 60\%\qquad
\textbf{(B)}\ 65\% \qquad
\textbf{(C)}\ 70\%\qquad
\textbf{(D)}\ 75\% \qquad
\textbf{(E)}\ 90\%$
2017 Czech And Slovak Olympiad III A, 3
Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$