Found problems: 85335
2019 PUMaC Geometry B, 4
Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?
2014 NIMO Summer Contest, 10
Among $100$ points in the plane, no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$.
[i]Proposed by Evan Chen[/i]
2018 CCA Math Bonanza, L4.3
$ABC$ is an isosceles triangle with $AB=AC$. Point $D$ is constructed on AB such that $\angle{BCD}=15^\circ$. Given that $BC=\sqrt{6}$ and $AD=1$, find the length of $CD$.
[i]2018 CCA Math Bonanza Lightning Round #4.3[/i]
2022 HMNT, 20
Let $\triangle{ABC}$ be an isosceles right triangle with $AB=AC=10.$ Let $M$ be the midpoint of $BC$ and $N$ the midpoint of $BM.$ Let $AN$ hit the circumcircle of $\triangle{ABC}$ again at $T.$ Compute the area of $\triangle{TBC}.$
2007 Hungary-Israel Binational, 3
Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.
2013 Saudi Arabia GMO TST, 4
In acute triangle $ABC$, points $D$ and $E$ are the feet of the perpendiculars from $A$ to $BC$ and $B$ to $CA$, respectively. Segment $AD$ is a diameter of circle $\omega$. Circle $\omega$ intersects sides $AC$ and $AB$ at $F$ and $G$ (other than $A$), respectively. Segment $BE$ intersects segments $GD$ and $GF$ at $X$ and $Y$ respectively. Ray $DY$ intersects side $AB$ at $Z$. Prove that lines $XZ$ and $BC$ are perpendicular
V Soros Olympiad 1998 - 99 (Russia), 8.5
Points $A$, $B$ and $C$ lie on one side of the angle with the vertex at point $O$, and points $A'$, $B'$ and $C'$ lie on the other. It is known that$ B$ is the midpoint of the segment $AC$, $B'$ is the midpoint of the segment $A'C'$, and lines $AA'$, $BB'$ and $CC'$ are parallel (fig.). Prove that the centers of the circles circumscribed around the triangles $OAC$, $OA'C$ and $OBB'$ lie on the same straight line.
[img]https://cdn.artofproblemsolving.com/attachments/d/6/92831077781bc45f25e9f71077034f84753a59.png[/img]
2021 Malaysia IMONST 2, 3
Given a sequence of positive integers
$$a_1, a_2, a_3, a_4, a_5, \dots$$
such that $a_2 > a_1$ and $a_{n+2} = 3a_{n+1} - 2a_n$ for all $n \geq 1$.
Prove that $a_{2021} > 2^{2019}$.
2022 MOAA, 13
Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$
.
Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.
1991 China Team Selection Test, 2
Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions:
(1) $f(0) = 0, f(1) = 1,$
(2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$
Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$
2004 Junior Balkan Team Selection Tests - Romania, 3
Let $A$ be a set of positive integers such that
a) if $a\in A$, the all the positive divisors of $a$ are also in $A$;
b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$.
Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.
2012 CIIM, Problem 2
A set $A\subset \mathbb{Z}$ is "padre" if whenever $x,y \in A$ with $x\leq y$ then also $2y -x \in A$. Prove that if $A$ is "padre", $0,a,b \in A$ with $0< a < b$ and $d = g.c.d(a,b)$ then \[a+b-3d, a+b-2d \in A.\]
2021 ELMO Problems, 2
Let $n > 1$ be an integer and let $a_1, a_2, \ldots, a_n$ be integers such that $n \mid a_i-i$ for all integers $1 \leq i \leq n$. Prove there exists an infinite sequence $b_1,b_2, \ldots$ such that
[list]
[*] $b_k\in\{a_1,a_2,\ldots, a_n\}$ for all positive integers $k$, and
[*] $\sum\limits_{k=1}^{\infty}\frac{b_k}{n^k}$ is an integer.
[/list]
2005 AIME Problems, 13
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.
2018 Moscow Mathematical Olympiad, 9
$x$ and $y$ are integer $5$-digits numbers, such that in the decimal notation, all ten digits are used exactly once. Also
$\tan{x}-\tan{y}=1+\tan{x}\tan{y}$, where $x,y$ are angles in degrees. Find maximum of $x$
2012 Junior Balkan Team Selection Tests - Moldova, 2
Let $ a,b,c $ be positive real numbers, prove the inequality:
$ (a+b+c)^2+ab+bc+ac\geq 6\sqrt{abc(a+b+c)} $
2017 Hanoi Open Mathematics Competitions, 10
Consider all words constituted by eight letters from $\{C ,H,M, O\}$. We arrange the words in an alphabet sequence.
Precisely, the first word is $CCCCCCCC$, the second one is $CCCCCCCH$, the third is $CCCCCCCM$, the fourth one is $CCCCCCCO, ...,$ and the last word is $OOOOOOOO$.
a) Determine the $2017$th word of the sequence?
b) What is the position of the word $HOMCHOMC$ in the sequence?
2010 Vietnam National Olympiad, 3
In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$
such that $BC$ not is the diameter.Consider a point $A$ varies in
$(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$
respective is intersect of $BC$ and internal and external bisector
of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through
orthocenter of $\triangle ABC$
and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.
1/Prove that $MN$ pass through a fixed point
2/Determint the place of $A$ such that $S_{AMN}$ has maxium value
2004 USAMTS Problems, 2
For the equation \[ (3x^2+y^2-4y-17)^3-(2x^2+2y^2-4y-6)^3=(x^2-y^2-11)^3, \]
determine its solutions $(x, y)$ where both $x$ and $y$ are integers. Prove that your answer lists all the integer solutions.
2010 Greece Team Selection Test, 1
Solve in positive reals the system:
$x+y+z+w=4$
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$
2009 VJIMC, Problem 1
A positive integer $m$ is called self-descriptive in base $b$, where $b\ge2$ is an integer, if
i) The representation of $m$ in base $b$ is of the form $(a_0a_1\ldots a_{b-1})_b$ (that is $m=a_0b^{b-1}+a_1b^{b-2}+\ldots+a_{b-2}b+a_{b-1}$, where $0\le a_i\le b-1$ are integers).
ii) $a_i$ is equal to the number of occurences of the number $i$ in the sequence $(a_0a_1\ldots a_{b-1})$.
For example, $(1210)_4$ is self-descriptive in base $4$, because it has four digits and contains one $0$, two $1$s, one $2$ and no $3$s.
1988 Balkan MO, 2
Find all polynomials of two variables $P(x,y)$ which satisfy
\[P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.\]
2022 Bulgaria National Olympiad, 6
Let $n\geq 2$ be a positive integer. The sets $A_{1},A_{2},\ldots, A_{n}$ and $B_{1},B_{2},\ldots, B_{n}$ of positive integers are such that $A_{i}\cap B_{j}$ is non-empty $\forall i,j\in\{1,2,\ldots ,n\}$ and $A_{i}\cap A_{j}=\o$, $B_{i}\cap B_{j}=\o$ $\forall i\neq j\in \{1,2,\ldots, n\}$. We put the elements of each set in a descending order and calculate the differences between consecutive elements in this new order. Find the least possible value of the greatest of all such differences.
2011 JBMO Shortlist, 6
Let $\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}$. Show that:$$\displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044}$$
When the equality holds?
2006 VJIMC, Problem 3
Two players play the following game: Let $n$ be a fixed integer greater than $1$. Starting from number $k=2$, each player has two possible moves: either replace the number $k$ by $k+1$ or by $2k$. The player who is forced to write a number greater than $n$ loses the game. Which player has a winning strategy for which $n$?