Found problems: 85335
The cells of an $n \times n$ chessboard are coloured in several colours so that no $2\times 2$ square contains four cells of the same colour. A [i]proper path [/i] of length $m$ is a sequence $a_1,a_2,..., a_m$ of distinct cells in which the cells $a_i$ and $a_{i+1}$ have a common side and are coloured in different colours for all $1 \le i < m$. Show that there exists a proper path of length $n$.
Let $ a,b,c$ be positive reals; show that $ \displaystyle a \plus{} b \plus{} c \leq \frac {bc}{b \plus{} c} \plus{} \frac {ca}{c \plus{} a} \plus{} \frac {ab}{a \plus{} b} \plus{} \frac {1}{2}\left(\frac {bc}{a} \plus{} \frac {ca}{b} \plus{} \frac {ab}{c}\right)$
[i](Darij Grinberg)[/i]
In an acute triangle $ABC$, points $D,E,F$ are the feet of the altitudes from $A,B,C$, respectively. A line through $D$ parallel to $EF$ meets $AC$ at $Q$ and $AB$ at $R$. Lines $BC$ and $EF$ intersect at $P$. Prove that the circumcircle of triangle $PQR$ passes through the midpoint of $BC$.
Given a real number $A$ and an integer $n$ with $2 \leq n \leq 19$, find all polynomials $P(x)$ with real coefficients such that $P(P(P(x))) = Ax^n +19x+99$.
Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$
The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.
Given a regular hexagon $ABCDEF$, let point $P$ be the intersection of lines $BC$ and $DE$, and let point $Q$ be the intersection of lines $AP$ and $CD$. If the area of $\triangle QEP$ is equal to $72$, find the area of regular hexagon $ABCDEF$.
[i]Proposed by [b]DeToasty3[/b][/i]
The center of the circumcircle of $\triangle ABC$ is $O$. The incenter of the triangle is $I$, and the intouch triangle is $A_1B_1C_1$. Let $H_1$ be the orthocenter of $\triangle A_1B_1C_1$. Prove that $O$, $I$, and $H_1$ are collinear.
A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance $d$ from it. By distance is meant the shortest distance measured over the curved survace of the lampshade. Prove that the area of the lampshade is $d^2(2\theta + \sqrt 3)$ where $\cot \frac {\theta}{2} = \frac{3}{\theta}.$
Consider the right isosceles $\vartriangle ABC$ at $ A$. Let $L$ be the intersection of the bisector of $\angle ACB$ with $AB$ and $K$ the intersection point of $CL$ with the bisector of $BC$. Let $X$ be the point on line $AK$ such that $\angle KCX = 90^o$ and let $Y$ be the point of intersection of $CX$ with the circumcircle of $\vartriangle ABC$. Let $Y'$ the reflection of point $Y$ wrt $BC$. Prove that $B - K -Y'$.
Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.
Let $y = mx + b$ be the image when the line $x - 3y + 11 = 0$ is reflected across the x-axis. The value of $m + b$ is
$ \textbf{(A)}\ -6\qquad\textbf{(B)}\ -5\qquad\textbf{(C)}\ -4\qquad\textbf{(D)}\ -3\qquad\textbf{(E)}\ -2 $
Let $ A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.
Let $n \ge k \ge 3$ be integers. Show that for every integer sequence $1 \le a_1 < a_2 < . . . <
a_k \le n$ one can choose non-negative integers $b_1, b_2, . . . , b_k$, satisfying the following conditions:
[list=i]
[*] $0 \le b_i \le n$ for each $1 \le i \le k$,
[*] all the positive $b_i$ are distinct,
[*] the sums $a_i + b_i$, $1 \le i \le k$, form a permutation of the first $k$ terms of a non-constant arithmetic
progression.
[/list]
It is known that for some $a$ and $b$ the equation $$\frac{x-3}{(x-6)^2} -\frac{x-6}{(x-3)^2} =a(b-9x+x^2)$$ has as its largest root the number $1995$. Find the smallest root of this equation for the same $a$ and $b$.
A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$.
Does there exist a sum-full zero-sum-free set of integers?
[i]Romania (Dan Schwarz)[/i]
Prove that for real $x\ge 1$, holds the inequality
$$\frac{2^x +3^x }{3^x +4^x} \le \frac57$$
Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.
Find all continuous functions $f:[1,8]\to\mathbb R$, such that
$$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$
Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.
Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$.
Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$.
Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.
Every two of the $n$ cities of Ruritania are connected by a direct flight of one from two airlines. Promonopoly Committee wants at least $k$ flights performed by one company. To do this, he can at least every day to choose any three cities and change the ownership of the three flights connecting these cities each other (that is, to take each of these flights from a company that performs it, and pass the other). What is the largest $k$ committee knowingly will be able to achieve its goal in no time, no matter how the flights are distributed hour?
Let $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$, and $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$ Determine $ a_{14^{14}}$.
A wheel consists of a fixed circular disk and a mobile circular ring. On the disk the numbers $1, 2, 3, \ldots ,N$ are marked, and on the ring $N$ integers $a_1,a_2,\ldots ,a_N$ of sum $1$ are marked. The ring can be turned into $N$ different positions in which the numbers on the disk and on the ring match each other. Multiply every number on the ring with the corresponding number on the disk and form the sum of $N$ products. In this way a sum is obtained for every position of the ring. Prove that the $N$ sums are different.
In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.
Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$.
$$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$
Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]