Found problems: 85335
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
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On the infinite chessboard several rectangular pieces are placed whose sides run along the grid lines. Each two have no squares in common, and each consists of an odd number of squares. Prove that these pieces can be painted in four colours such that two pieces painted in the same colour do not share any boundary points.
Let $ n \geq 2$ be a positive integer and $ \lambda$ a positive real number. Initially there are $ n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $ A$ and $ B$, with $ A$ to the left of $ B$, and letting the flea from $ A$ jump over the flea from $ B$ to the point $ C$ so that $ \frac {BC}{AB} \equal{} \lambda$.
Determine all values of $ \lambda$ such that, for any point $ M$ on the line and for any initial position of the $ n$ fleas, there exists a sequence of moves that will take them all to the position right of $ M$.
Let $x$ be a real number with the following property: for each positive integer $q$, there exists an integer $p$, such that
\[\left|x-\frac{p}{q} \right|<\frac{1}{3q}. \]
Prove that $x$ is an integer.
Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that:
[b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color,
and
[b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$.
[i]Author: Gerhard Wöginger, Netherlands[/i]
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i]
Let $x_0, x_1, x_2, \ldots$ be the sequence defined by
$x_i= 2^i$ if $0 \leq i \leq 2003$
$x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.
Points $C_1, B_1$ on sides $AB, AC$ respectively of triangle $ABC$ are such that $BB_1 \perp CC_1$. Point $X$ lying inside the triangle is such that $\angle XBC = \angle B_1BA, \angle XCB = \angle C_1CA$. Prove that $\angle B_1XC_1 =90^o- \angle A$.
(A. Antropov, A. Yakubov)
a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$.
b)Let $a>0$. Prove that:
\[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\]
[i]***[/i]
Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.
Jerry writes down all binary strings of length $10$ without any two consecutive $1$s. How many $1$s does Jerry write?
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
Let $A=\{1,2,...,100\}$ and $f(k), k\in N$ be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k)=50$?
Suppose that $ M$ is an arbitrary point on side $ BC$ of triangle $ ABC$. $ B_1,C_1$ are points on $ AB,AC$ such that $ MB = MB_1$ and $ MC = MC_1$. Suppose that $ H,I$ are orthocenter of triangle $ ABC$ and incenter of triangle $ MB_1C_1$. Prove that $ A,B_1,H,I,C_1$ lie on a circle.
All sides of the convex pentagon $ ABCDE$ are of equal length, and $ \angle A \equal{} \angle B \equal{} 90^{\circ}$. What is the degree measure of $ \angle E$?
$ \textbf{(A)}\ 90 \qquad \textbf{(B)}\ 108 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$
Let \(P\) be a convex polyhedron with the following properties:
[b]1)[/b] \(P\) has exactly \(666\) edges.
[b]2)[/b] The degrees of all vertices of \(P\) differ by at most \(1\).
[b]3)[/b] There is an edge‐coloring of \(P\) with \(k\) colors such that for each color \(c\) and any two distinct vertices \(V_1,V_2\), there exists a path from \(V_1\) to \(V_2\) all of whose edges have color \(c\).
Determine the largest positive integer \(k\) for which such a polyhedron \(P\) exists.
There is a king in the lower left corner of a chessboard of dimensions $6$ and $6$. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different paths can the king take to the upper right corner of the board?
The real numbers $x,y,z$ satisfy $x^2+y^2+z^2=3.$ Prove that the inequality
$x^3-(y^2+yz+z^2)x+yz(y+z)\le 3\sqrt{3}.$
and find all triples $(x,y,z)$ for which equality holds.
Show that for all real numbers $x$ and $y$ with $x > -1$ and $y > -1$ and $x + y = 1$ the inequality
$$\frac{x}{y + 1} +\frac{y}{x + 1} \ge \frac23$$
holds. When does equality apply?
[i](Walther Janous)[/i]
Let $ABC$ be an isosceles triangle with $AC = BC$, and let $M$ be the midpoint of $AB$. Let $P$ be a point inside the triangle such that $\angle PAB =\angle PBC$. Prove that $\angle APM + \angle BPC = 180^o$.
A $7\times 7$ board has a lamp on each of its $49$ squares, which can be on or off.
The allowed operation is to choose $3$ consecutive cells of a row or a column that have two lamps neighboring each other on and the other off, and change the state of all three. Namely
[img]https://cdn.artofproblemsolving.com/attachments/e/b/28737b19c940ff5e1c98d05533c77069e990f5.png[/img]
Give a configuration of exactly $8$ lit lamps located in the first $4$ rows of the board such that, through a succession of permitted operations, a single lamp is lit on the board and that it is located in the last row. Show the sequence of operations used to achieve the goal.
If $m=2+2\sqrt{44n^2+1}$ is an integer then show that it is also a perfect square. Here $n$ is a natural number.
([b]4[/b]) Find all $ y > 1$ satisfying $ \int^y_1x\ln x\ dx \equal{} \frac {1}{4}$.
Let $ABC$ be an acute triangle, which is not equilateral. Denote by $O$ and $H$ its circumcenter and orthocenter, respectively. The circle $k$ passes through $B$ and touches the line $AC$ at $A$. The circle $l$ with center on the ray $BH$ touhes the line $AB$ at $A$. The circles $k$ and $l$ meet in $X$ ($X\ne A$). Show that $\angle HXO=180^\circ-\angle BAC$.
[i]Proposed by Josef Tkadlec[/i]
For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.