Found problems: 85335
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?
2009 AMC 12/AHSME, 22
A regular octahedron has side length $ 1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $ \frac {a\sqrt {b}}{c}$, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ b$ is not divisible by the square of any prime. What is $ a \plus{} b \plus{} c$?
$ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14$
2001 Mediterranean Mathematics Olympiad, 4
Let $S$ be the set of points inside a given equilateral triangle $ABC$ with side $1$ or on its boundary. For any $M \in S, a_M, b_M, c_M$ denote the distances from $M$ to $BC,CA,AB$, respectively. Define
\[f(M) = a_M^3 (b_M - c_M) + b_M^3(c_M - a_M) + c_M^3(a_M - b_M).\]
[b](a)[/b] Describe the set $\{M \in S | f(M) \geq 0\}$ geometrically.
[b](b)[/b] Find the minimum and maximum values of $f(M)$ as well as the points in which these are attained.
2019 Irish Math Olympiad, 7
Three non-zero real numbers $a, b, c$ satisfy $a + b + c = 0$ and $a^4 + b^4 + c^4 = 128$. Determine all possible values of $ab + bc + ca$.
2022 USA TSTST, 3
Determine all positive integers $N$ for which there exists a strictly increasing sequence of positive integers $s_0<s_1<s_2<\cdots$ satisfying the following properties:
[list=disc]
[*]the sequence $s_1-s_0$, $s_2-s_1$, $s_3-s_2$, $\ldots$ is periodic; and
[*]$s_{s_n}-s_{s_{n-1}}\leq N<s_{1+s_n}-s_{s_{n-1}}$ for all positive integers $n$
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