Found problems: 85335
2006 All-Russian Olympiad Regional Round, 9.3
It is known that $x^2_1+ x^2_2+...+ x^2_6= 6$ and $x_1 + x_2 +....+ x_6 = 0.$ Prove that $ x_1x_2....x_6 \le \frac12$ .
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2024 Regional Competition For Advanced Students, 4
Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$.
[i](Walther Janous)[/i]
2021 NICE Olympiad, 4
Find all real numbers $c$ for which there exists a nonconstant two-variable polynomial $P(x, y)$ with real coefficients satisfying
\[[P(x, y)]^2 = P(cxy, x^2 + y^2)\]
for all real $x$ and $y$.
[i]Nikolai Beluhov and Konstantin Garov[/i]
Kyiv City MO 1984-93 - geometry, 1990.11.1
Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.
2021 Alibaba Global Math Competition, 4
Let $(\Omega, \mathcal{A},\mathbb{P})$ be a standard probability space, and $\mathcal{X}$ be the set of all bounded random variables. For $t>0$, defined the mapping $R_t$ by
\[R_t(X)=t\log \mathbb{E}[\exp(X/t)], \quad X \in \mathcal{X},\]
and for $t \in (0,1)$ define the mapping $Q_t$ by
\[Q_t(X)=\inf\{x \in \mathbb{R}: \mathbb{P}(X>x) \le t\}, \quad X \in \mathcal{X}.\]
For two mappings $f,g:\mathcal{X} \to [-\infty,\infty)$, defined the operator $\square$ by
\[f\square g(X)=\inf\{f(Y)+g(X-Y): Y \in \mathcal{X}\}, \quad X \in \mathcal{X}.\]
(a) Show that, for $t,s>0$,
\[R_t \square R_s=R_{t+s}.\]
(b) Show that, for $t,s>0$ with $t+s<1$,
\[Q_t \square Q_s=Q_{t+s}.\]
MathLinks Contest 7th, 4.1
Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that
\[ AB\plus{}BC\plus{}CA \plus{} DE < AD \plus{} AE \plus{} BD\plus{}BE \plus{} CD\plus{}CE .\]
2024 Harvard-MIT Mathematics Tournament, 24
A circle is tangent to both branches of the hyperbola $x^2 - 20y^2 = 24$ as well as the $x$-axis. Compute the area of this circle.
1982 Swedish Mathematical Competition, 1
How many solutions does
\[
x^2 - [x^2] = \left(x - [x]\right)^2
\]
have satisfying $1 \leq x \leq n$?
2010 Contests, 1
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
2011 Chile National Olympiad, 2
Let $O$ be the center of the circle circumscribed to triangle $ABC$ and let $ S_ {A} $, $ S_ {B} $, $ S_ {C} $ be the circles centered on $O$ that are tangent to the sides $BC, CA, AB$ respectively. Show that the sum of the angle between the two tangents $ S_ {A} $ from $A$ plus the angle between the two tangents $ S_ {B} $ from $B$ plus the angle between the two tangents $ S_ {C} $ from $C$ is $180$ degrees.
2009 Brazil Team Selection Test, 4
Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if
\[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\]
Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$.
[i]Proposed by Andrey Badzyan, Russia[/i]
2023 Kyiv City MO Round 1, Problem 5
You are given a square $n \times n$. The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$, find the largest value of $m$ for which it is possible.
[i]Proposed by Oleksiy Masalitin, Fedir Yudin[/i]
1993 IberoAmerican, 1
Let $ABC$ be an equilateral triangle and $\Gamma$ its incircle. If $D$ and $E$ are points on the segments $AB$ and $AC$ such that $DE$ is tangent to $\Gamma$, show that $\frac{AD}{DB}+\frac{AE}{EC}=1$.
2022 Regional Olympiad of Mexico West, 1
Find a subset of $\{1,2, ...,2022\}$ with maximum number of elements such that it does not have two elements $a$ and $b$ such that $a = b + d$ for some divisor $d$ of $b$.
2021 Adygea Teachers' Geometry Olympiad, 3
In a triangle, one excircle touches side $AB$ at point $C_1$ and the other touches side $BC$ at point $A_1$. Prove that on the straight line $A_1C_1$ the constructed excircles cut out equal segments.
2023-IMOC, A5
We can conduct the following moves to a real number $x$: choose a positive integer $n$, and positives reals $a_1,a_2,\cdots, a_n$ whose reciprocals sum up to $1$. Let $x_0=x$, and $x_k=\sqrt{x_{k-1}a_k}$ for all $1\leq k\leq n$. Finally, let $y=x_n$. We said $M>0$ is "tremendous" if for any $x\in \mathbb{R}^+$, we can always choose $n,a_1,a_2,\cdots, a_n$ to make the resulting $y$ smaller than $M$. Find all tremendous numbers.
[i]Proposed by ckliao914.[/i]
2000 Italy TST, 1
Determine all triples $(x,y,z)$ of positive integers such that
\[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]
1997 National High School Mathematics League, 8
Line $l$ that passes right focal point of hyperbola $x^2-\frac{y^2}{2}=1$ intersects the hyperbola at $A,B$. The number of line $l$ that $|AB|=\lambda$ is 3, then $\lambda=$________.
2020 Tuymaada Olympiad, 4
For each positive integer $k$, let $g(k)$ be the maximum possible number of points in the plane such that pairwise distances between these points have only $k$ different values. Prove that there exists $k$ such that $g(k) > 2k + 2020$.
2016 Math Prize for Girls Problems, 16
Let $A < B < C < D$ be positive integers such that every three of them form the side lengths of an obtuse triangle. Compute the least possible value of $D$.
1967 IMO Shortlist, 2
Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?
2007 Iran MO (2nd Round), 2
Tow circles $C,D$ are exterior tangent to each other at point $P$. Point $A$ is in the circle $C$. We draw $2$ tangents $AM,AN$ from $A$ to the circle $D$ ($M,N$ are the tangency points.). The second meet points of $AM,AN$ with $C$ are $E,F$, respectively. Prove that $\frac{PE}{PF}=\frac{ME}{NF}$.
2010 Baltic Way, 13
In an acute triangle $ABC$, the segment $CD$ is an altitude and $H$ is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle $DHB$, determine all possible values of $\angle CAB$.
2017 Azerbaijan JBMO TST, 1
a,b,c>0 and $abc\ge 1$.Prove that:
$\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$
2014 Contests, 3
Four different positive integers less than 10 are chosen randomly. What is the probability that their sum is odd?