Found problems: 85335
$9$ different pieces of cheese are placed on a plate. Is it always possible to cut one of them into two parts so that the $10$ pieces obtained were divisible into two portions of equal mass of $5$ pieces each?
What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?
It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Can it happen that from segments of lengths$$\sqrt{a^2 + bc}, \quad \sqrt{b^2 + ca}, \quad \sqrt{c^2 + ab}$$an obtuse triangle can be formed?
For each positive integer $n$, let $H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}$. If \[ \sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{M}{N} \] for relatively prime positive integers $M$ and $N$, compute $100M+N$.
[i]Based on a proposal by ssilwa[/i]
Color every vertex of $2008$-gon with two colors, such that adjacent vertices have different color. If sum of angles of vertices of first color is same as sum of angles of vertices of second color, than we call $2008$-gon as interesting.
Convex $2009$-gon one vertex is marked. It is known, that if remove any unmarked vertex, then we get interesting $2008$-gon. Prove, that if we remove marked vertex, then we get interesting $2008$-gon too.
Given an acute-angled triangle $ABC$ in which $|AB| <|AC|$. Point $D$ is the midpoint of the shorter arc $BC$ of its circumcircle. The point $I$ is the center of its incircle, and the point $J$ is symmetric point of $I$ wrt line $BC$. The line $DJ$ intersects the circumcircle of the triangle $ABC$ at the point $E$ belonging to the arc $AB$. Prove that $|AI |= |IE|$.
Find all values of $n$ for which there exists a convex cyclic non-regular polygon with $n$ vertices such that the measures of all its internal angles are equal.
The triangle $ABC$ has $CA = CB$. $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $AB$ to $C$). $D$ is the foot of the perpendicular from $C$ to $PB$. Show that $PA + PB = 2 \cdot PD$.
Let $n \geq 3$ be an integer. Find the minimum degree of one algebraic (polynomial) equation that defines the set of vertices of the correct $n$-gon on plane $R^2$.
Give the positive root(s) of $ x^3 \plus{} 2x^2 \minus{} 2x \minus{} 4$.
Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.
Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.
Let $a, b, c, \lambda$ be positive real numbers with $\lambda \geq 1/4$. Show that $$\frac{a}{\sqrt{b^2+\lambda bc+c^2}}+\frac{b}{\sqrt{c^2+\lambda ca+a^2}}+\frac{c}{\sqrt{a^2+\lambda ab+b^2}} \geq \frac{3}{\sqrt{\lambda +2}}.$$
$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.
Let $ A$, $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \plus{} B \equal{} ?$
$\textbf{(A)}\ 1998 \qquad\textbf{(B)}\ 1999 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ 3998$
There are $2000$ cities in the country, some pairs of cities are connected by roads. It is known that no more than $N$ different non-self-intersecting cyclic routes of odd length. Prove that the country can be divided into $N + 2$ republics so that no two cities from the same republic are connected by a road.
Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied:
a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$.
b) For any $x \in M$, one has that $-x \in M$.
c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$.
For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.
A triangle has two sides of lengths $1984$ and $2016$. Find the maximum possible area of the triangle.
[i]Proposed by Nathan Ramesh
Let $g$ be a straight line and $n$ a given positive integer. Prove that there are always n different points on g to choose as well as a point not lying on g in such a way that the distance between each two of these $n + 1$ points is an integer.
Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:
\[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3}
\]
In what case does equality hold?
Let $a_1\in(0,1)$ and define recursively the sequence $(a_n)_{n\geq 1}$ by $a_{n+1}=3a_n-4a_n^3$ for all $n\geq 1$.
a) Prove that for all $n$ we have $|a_n|<1$.
b) Prove that for any $k\geq 2$ we can choose $a_1\in(0,1)$ adequately such that $a_{n+k}=a_n$ for all $n\geq 1$.
[i]Sergiu Moroianu[/i]
Let $ZHAO$ be a square with area $2024$. Let $X$ be the center of this square and let $C$, $D$, $E$, $K$ be the centroids of $XZH$, $XHA$, $XAO$, and $XOZ$, respectively. Find $[ZHAO]$ $+$ $[CZHAO]$ $+$ $[DZHAO]$ $+$ $[EZHAO]$ $+$ $[KZHAO]$.
(Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.)
Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
Given that the answer to this problem can be expressed as $a\cdot b\cdot c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers with $b=10$, compute $1000a+100b+10c$.
[i]Proposed by Ankit Bisain[/i]