Found problems: 85335
2001 Tuymaada Olympiad, 2
Non-zero numbers are arranged in $n \times n$ square ($n>2$). Every number is exactly $k$ times less than the sum of all the other numbers in the same cross (i.e., $2n-2$ numbers written in the same row or column with this number).
Find all possible $k$.
[i]Proposed by D. Rostovsky, A. Khrabrov, S. Berlov [/i]
2006 Germany Team Selection Test, 1
Let $ ABC$ be an equilateral triangle, and $ P,Q,R$ three points in its interior satisfying
\[ \measuredangle PCA \equal{} \measuredangle CAR \equal{} 15^{\circ},\ \measuredangle RBC \equal{} \measuredangle BCQ \equal{} 20^{\circ},\ \measuredangle QAB \equal{} \measuredangle ABP \equal{} 25^{\circ}.\] Compute the angles of triangle $ PQR$.
1984 IMO Longlists, 53
Find a sequence of natural numbers $a_i$ such that $a_i = \displaystyle\sum_{r=1}^{i+4} d_r$, where $d_r \neq d_s$ for $r \neq s$ and $d_r$ divides $a_i$.
1999 IMO, 2
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality
\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]
holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.
2007 Princeton University Math Competition, 4
A positive integer is called [i]squarefree[/i] if its only perfect square factor is $1$. Call a set of positive integers [i]squarefreeful[/i] if each product of two of its elements is squarefree, and [i]squarefreefullest[/i] if no positive integer less than the maximum element of the set can be added while preserving the set's squarefreefulness. What is the minimum number of elements in a squarefreefullest set containing $31$?
2009 Kosovo National Mathematical Olympiad, 4
Prove that if in the product of four consequtive natural numbers we add $1$, we get a perfect square.
2013 Harvard-MIT Mathematics Tournament, 3
Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers. Determine the $100$th smallest element of $S$.
2018 Spain Mathematical Olympiad, 3
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$.
Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.
2020 Kürschák Competition, P1
Let $n$ and $k$ be positive integers. Given $n$ closed discs in the plane such that no matter how we choose $k + 1$ of them, there are always two of the chosen discs that have no common point. Prove that the $n$ discs can be partitioned into at most $10k$ classes such that any two discs in the same class have no common point.
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 $1976.$