Found problems: 85335
Consider an odd prime number $p$ and $p$ consecutive positive integers $m_1,m_2,…,m_p$.
Choose a permutation $\sigma$ of $1,2,…,p$ .
Show that there exist two different numbers $k,l\in{(1,2,…,p)}$ such that
$p\mid{m_k.m_{\sigma(k)}-m_l.m_{\sigma(l)}}$
The chromatic number $\chi$ of an (infinite) plane is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart.
Prove that $4 \leq \chi \leq 7$.
Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.
[i]Proposed by Horst Sewerin, Germany[/i]
The vertices of a triangle have integer coordinates and one of its sides is of length $\sqrt{n}$, where $n$ is a square-free natural number. Prove that the ratio of the circumradius and the inradius is an irrational number.
Given a circle and a quadrilateral $ABCD$ whose vertices all lie on the circle. Let $R$ be the midpoint of arc $AB$. The line $RC$ meets line $AB$ at point $S$, and the line $RD$ meets line $AB$ at point $T$.
Prove that $CDTS$ is a cyclic quadrilateral.
The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off of each of the mirrors once and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\tfrac{AB}{AX} = 64$, $\tfrac{CA}{AY} = \tfrac pq$ for coprime positive integers $p$, $q$. Find $p + q$.
[i]Proposed by Sid Doppalapudi[/i]
Construct an uncountable Hausdorff space in which the complement of the closure of any nonempty, open set is countable.
[i]A. Hajnal, I. Juhasz[/i]
Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.
$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.
Solve the following system of equations in integers:
$\begin{cases}
x^2+2xy+8z=4z^2+4y+8\\
x^2+y+2z=156 \\
\end{cases}$
The sequence $(s_n)$, where $s_n= \sum_{k=1}^n \sin k$ for $n = 1, 2,\dots$ is bounded. Find an upper and lower bound.
Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$ of a triangle $ABC$. It is known that $\angle MAN = 15^o$ and $\angle BAN = 45^o$. Find the value of angle $\angle ABM$.
(A. Blinkov)
What conditions must the real numbers $ a $, $ b $, and $ c $ satisfy so that the equation
$$ x^3 + ax^2 + bx + c = 0$$
has three distinct real roots forming a geometric progression?
a conic with apotheme 1 slides (varying height and radius, with $r < \frac12$) so that the conic's area is $9$ times that of its inscribed sphere. What's the height of that conic?
A wedding is going to be held in a city with $25$ types of meals, to which some of the $2017$ citizens will be invited. All of the citizens like some meals and each meal is liked by at least one person. A "$suitable$ $list$" is a set of citizens, such that each meal is liked by at least one person in the set. A "$kamber$ $group$" is a set that contains at least one person from each "$suitable$ $list$". Given a "$kamber$ $group$", which has no subset (other than itself) that is also a "$kamber$ $group$", prove that there exists a meal, which is liked by everyone in the group.
$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$).
Prove that $l_1 || l_2$.
A necklace contains $2016$ pearls, each of which has one of the colours black, green or blue.
In each step we replace simultaneously each pearl with a new pearl, where the colour of the new pearl is determined as follows: If the two original neighbours were of the same colour, the new pearl has their colour. If the neighbours had two different colours, the new pearl has the third colour.
(a) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if half of the pearls were black and half of the pearls were green at the start?
(b) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if thousand of the pearls were black at the start and the rest green?
(c) Is it possible to transform a necklace that contains exactly two adjacent black pearls and $2014$ blue pearls to a necklace that contains one green pearl and $2015$ blue pearls?
Proposed byTheresia Eisenkölbl
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. The tangent line of the circumcircle of triangle $BHC$ at $H$ meets $AB$ and $AC$ at $E$ and $F$ respectively. If $O$ is the circumcenter of triangle $AEF$, prove that the circumcircle of triangle $EOF$ is tangent to $\Omega$.
[color=darkblue]Points $ X$, $ Y$ and $ Z$ are situated on the sides $ (BC)$, $ (CA)$ and $ (AB)$ of the triangles $ ABC$, such that triangles $ XYZ$ and $ ABC$ are similiar. Prove that circumcircle of $ AYZ$ passes through a fixed point.[/color]
Find the value of $a$ satisfying
\begin{align*}
a+b&=3\\
b+c&=11\\
c+a&=61
\end{align*}
Let $ABCD$ be a trapezoid with $AB$ parallel to $CD, AB$ has length $1,$ and $CD$ has length $41.$ Let points $X$ and $Y$ lie on sides $AD$ and $BC,$ respectively, such that $XY$ is parallel to $AB$ and $CD,$ and $XY$ has length $31.$ Let $m$ and $n$ be two relatively prime positive integers such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $\tfrac{m}{n}.$ Find $m+2n.$
Let $\gamma_A, \gamma_B, \gamma_C$ be excircles of triangle $ABC$, touching the sides $BC$, $CA$, $AB$ respectively. Let $l_A$ denote the common external tangent to $\gamma_B$ and $\gamma_C$ distinct from $BC$. Define $l_B, l_C$ similarly. The tangent from a point $P$ of $l_A$ to $\gamma_B$ distinct from $l_A$ meets $l_C$ at point $X$. Similarly the tangent from $P$ to $\gamma_C$ meets $l_B$ at $Y$. Prove that $XY$ touches $\gamma_A$.
You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?
A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$.