Found problems: 85335
MBMT Team Rounds, 2015 F13 E11
Two (not necessarily different) integers between $1$ and $60$, inclusive, are chosen independently and at random. What is the probability that their product is a multiple of $60$?
2022 HMNT, 6
A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected without either player winning, the game ends in a draw. Given that both players play optimally, find all positive areas of the convex polygon formed by Alice's points at the end of the game.
2022 Moscow Mathematical Olympiad, 4
The starship is in a half-space at a distance $a$ from its boundary. The crew knows about it, but has no idea in which direction to move in order to reach the boundary plane. The starship can fly in space along any trajectory, measuring the length of the path traveled, and has a sensor that sends a signal when
the border has been reached. Can a starship be guaranteed to reach the border with a path no longer than $14a$?
2022 Indonesia MO, 6
In a triangle $ABC$, $D$ and $E$ lies on $AB$ and $AC$ such that $DE$ is parallel to $BC$. There exists point $P$ in the interior of $BDEC$ such that
\[ \angle BPD = \angle CPE = 90^{\circ} \]Prove that the line $AP$ passes through the circumcenter of triangles $EPD$ and $BPC$.
2006 China Team Selection Test, 1
Let the intersections of $\odot O_1$ and $\odot O_2$ be $A$ and $B$. Point $R$ is on arc $AB$ of $\odot O_1$ and $T$ is on arc $AB$ on $\odot O_2$. $AR$ and $BR$ meet $\odot O_2$ at $C$ and $D$; $AT$ and $BT$ meet $\odot O_1$ at $Q$ and $P$. If $PR$ and $TD$ meet at $E$ and $QR$ and $TC$ meet at $F$, then prove: $AE \cdot BT \cdot BR = BF \cdot AT \cdot AR$.
2022 IMO Shortlist, A4
Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that
\[2^{j-i}x_ix_j>2^{s-3}.\]
1966 IMO Shortlist, 40
For a positive real number $p$, find all real solutions to the equation
\[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]
1982 Poland - Second Round, 1
Prove that if $ c, d $ are integers with $ c \neq d $, $ d > 0 $ then the equation $$ x^3 - 3cx^2 - dx + c = 0$$
has no more than one rational root.
2021 Federal Competition For Advanced Students, P1, 1
Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$
[size=50](Marian Dinca)[/size]
2024 Mexico National Olympiad, 3
Let $ABCDEF$ a convex hexagon, and let $A_1,B_1,C_1,D_1,E_1$ y $F_1$ be the midpoints of $AB,BC,CD,$ $DE,EF$ and $FA$, respectively. Construct points $A_2,B_2,C_2,D_2,E_2$ and $F_2$ in the interior of $A_1B_1C_1D_1E_1F_1$ such that both
1. The sides of the dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ are all equal and
2. $\angle A_1B_2B_1+\angle C_1D_2D_1+\angle E_1F_2F_1=\angle B_1C_2C_1+\angle D_1E_2E_1+\angle F_1A_2A_1=360^\circ$, where all these angles are less than $180 ^\circ$,
Prove that $A_2B_2C_2D_2E_2F_2$ is cyclic.
[b]Note:[/b] Dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ is shaped like a 6-pointed star, where the points are $A_1,B_1,C_1,D_1,E_1$ y $F_1$.
2019 Czech and Slovak Olympiad III A, 1
Find all triplets $(x,y,z)\in\mathbb{R}^3$ such that
\begin{align*}
x^2-yz &= |y-z|+1, \\
y^2-zx &= |z-x|+1, \\
z^2-xy &= |x-y|+1.
\end{align*}
2007 Mexico National Olympiad, 3
Given $a$, $b$, and $c$ be positive real numbers with $a+b+c=1$, prove that
\[\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2\]
2013 Gheorghe Vranceanu, 1
Find both extrema of the function $ x\to\frac{\sin x-3}{\cos x +2} .$
2023/2024 Tournament of Towns, 3
3. Let us call a bi-squared card $2 \times 1$ regular, if two positive integers are written on it and the number in the upper square is less than the number in the lower square. It is allowed at each move to change both numbers in the following manner: either add the same integer (possibly negative) to both numbers, or multiply each number by the same positive integer, or divide each number by the same positive integer. The card must remain regular after any changes made. What minimal number of moves is sufficient to get any regular card from any other regular card?
Alexey Glebov
VI Soros Olympiad 1999 - 2000 (Russia), 9.2
Solve the equation $[x]\{x\} = 1999x$, where $[x]$ denotes the largest integer less than or equal to $x$, and $\{x\} = x -[x] $
1976 IMO, 1
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
2004 Bulgaria Team Selection Test, 1
The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.
2023 UMD Math Competition Part I, #10
There are $100$ people in a room. Some are [i]wise[/i] and some are [i]optimists[/i].
$\quad \bullet~$ A [i]wise[/i] person can look at someone and know if they are wise or if they are an optimist.
$\quad \bullet~$ An [i]optimist[/i] thinks everyone is wise (including themselves).
Everyone in the room writes down what they think is the number of wise people in the room. What is the smallest possible value for the average?
$$
\mathrm a. ~ 10\qquad \mathrm b.~25\qquad \mathrm c. ~50 \qquad \mathrm d. ~75 \qquad \mathrm e. ~100
$$
1988 IMO Longlists, 71
The quadrilateral $A_1A_2A_3A_4$ is cyclic, and its sides are $a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4$ and $a_4 = A_4A_1.$ The respective circles with centres $I_i$ and radii $r_i$ are tangent externally to each side $a_i$ and to the sides $a_{i+1}$ and $a_{i-1}$ extended. ($a_0 = a_4$). Show that \[ \prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2. \]
2009 India National Olympiad, 3
Find all real numbers $ x$ such that:
$ [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x]$
(Here $ [x]$ denotes the largest integer not exceeding $ x$.)
2014 Iran Team Selection Test, 5
Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$, call a real number $a$ [b]good[/b] if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$. Also a subset of $X$ is called [b]compact[/b] if the average of its elements is a good number.
Prove that at least $2^{n-3}$ subsets of $X$ are compact.
[i]Proposed by Mahyar Sefidgaran[/i]
2008 Harvard-MIT Mathematics Tournament, 9
Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.
2016 BAMO, 4
Find a positive integer $N$ and $a_1, a_2, \cdots, a_N$ where $a_k = 1$ or $a_k = -1$, for each $k=1,2,\cdots,N,$ such that $$a_1 \cdot 1^3 + a_2 \cdot 2^3 + a_3 \cdot 3^3 \cdots + a_N \cdot N^3 = 20162016$$
or show that this is impossible.
2024 Mexico National Olympiad, 5
Let $A$ and $B$ infinite sets of positive real numbers such that:
1. For any pair of elements $u \ge v$ in $A$, it follows that $u+v$ is an element of $B$.
2. For any pair of elements $s>t$ in $B$, it follows that $s-t$ is an element of $A$.
Prove that $A=B$ or there exists a real number $r$ such that $B=\{2r, 3r, 4r, 5r, \dots\}$.
1994 All-Russian Olympiad, 7
A trapezoid $ABCD$ ($AB ///CD$) has the property that there are points $P$ and $Q$ on sides $AD$ and $BC$ respectively such that $\angle APB = \angle CPD$ and $\angle AQB = \angle CQD$. Show that the points $P$ and $Q$ are equidistant from the intersection point of the diagonals of the trapezoid.
(M. Smurov)