Let $ABC$ be a triangle such that $AB\neq AC$. We denote its orthocentre by $H$, its circumcentre by $O$ and the midpoint of $BC$ by $D$. The extensions of $HD$ and $AO$ meet in $P$. Prove that triangles $AHP$ and $ABC$ have the same centroid.

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.

Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a$ has a root on unit circle

If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{a}{b}$, to the nearest integer, is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{none of these} $

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

In triangle $ABC$, $\angle B=2\angle C$ and $\angle A>90^\circ$. Let $D$ be the point on the line $AB$ such that $CD$ is perpendicular to $AC$, and let $M$ be the midpoint of $BC$. Prove that $\angle AMB=\angle DMC$.

In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded) \[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \] as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.

The extensions of the sides $AB$ and $CD$ of the trapezoid $ABCD$ intersect at point $E$. Denote by $H$ and $G$ the midpoints of $BD$ and $AC$. Find the ratio of the area $AEGH$ to the area $ABCD$.

For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that: \[ \frac{AB}{AF}\times DC\plus{}\frac{AC}{AE}\times DB\equal{}\frac{AD}{AP}\times BC\]

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$.

Two cyclists, $ k$ miles apart, and starting at the same time, would be together in $ r$ hours if they traveled in the same direction, but would pass each other in $ t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is: $ \textbf{(A)}\ \frac {r \plus{} t}{r \minus{} t} \qquad\textbf{(B)}\ \frac {r}{r \minus{} t} \qquad\textbf{(C)}\ \frac {r \plus{} t}{r} \qquad\textbf{(D)}\ \frac {r}{t} \qquad\textbf{(E)}\ \frac {r \plus{} k}{t \minus{} k}$

Determine the locus of points, for whom the ratio of the distances to two given points has a constant value.

Let the incircle of the triangle $ABC$ touch $[BC]$ at $D$ and $I$ be the incenter of the triangle. Let $T$ be midpoint of $[ID]$. Let the perpendicular from $I$ to $AD$ meet $AB$ and $AC$ at $K$ and $L$, respectively. Let the perpendicular from $T$ to $AD$ meet $AB$ and $AC$ at $M$ and $N$, respectively. Show that $|KM|\cdot |LN|=|BM|\cdot|CN|$.

Given that $r$ and $s$ are relatively prime positive integers such that $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$, find $r$ and $s$.

A circle is inscribed in the trapezoid [i]ABCD[/i]. Let [i]K, L, M, N[/i] be the points of tangency of this circle with the diagonals [i]AC[/i] and [i]BD[/i], respectively ([i]K[/i] is between [i]A[/i] and [i]L[/i], and [i]M[/i] is between [i]B[/i] and [i]N[/i]). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, find the radius of the circle. [color=red][Moderator edit: A solution of this problem can be found on http://www.ajorza.org/math/mathfiles/scans/belarus.pdf , page 20 (the statement of the problem is on page 6). The author of the problem is I. Voronovich.][/color]

In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of edges $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ [asy] size(6cm); pair A,B,C,D,EE,F,G,H,I,J; C = (0,0); B = (-1,1); D = (2,0.5); A = B+D; G = (0,2); F = B+G; H = G+D; EE = G+B+D; I = (D+H)/2; J = (B+F)/2; filldraw(C--I--EE--J--cycle,lightgray,black); draw(C--D--H--EE--F--B--cycle); draw(G--F--G--C--G--H); draw(A--B,dashed); draw(A--EE,dashed); draw(A--D,dashed); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(I); dot(J); label("$A$",A,E); label("$B$",B,W); label("$C$",C,S); label("$D$",D,E); label("$E$",EE,N); label("$F$",F,W); label("$G$",G,N); label("$H$",H,E); label("$I$",I,E); label("$J$",J,W); [/asy] $\textbf{(A) } \frac{5}{4} \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{25}{16} \qquad \textbf{(E) } \frac{9}{4}$

At one moment, a kid noticed that the end of the hour hand, the end of the minute hand and one of the twelve numbers (regarded as a point) of his watch formed an equilateral triangle. He also calculated that $t$ hours would elapse for the next similar case. Suppose that the ratio of the lengths of the minute hand (whose length is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is $k>1$. Find the maximal value of $t$.

When the base of a triangle is increased $10\%$ and the altitude to this base is decreased $10\%$, the change in area is $\text{(A)} \ 1\%~ \text{increase} \qquad \text{(B)} \ \frac12 \%~ \text{increase} \qquad \text{(C)} \ 0\% \qquad \text{(D)} \ \frac12 \% ~\text{decrease} \qquad \text{(E)} \ 1\% ~\text{decrease}$

Let $ ABC$ be a triangle and $ AB\ne AC$ . $ D$ is a point on $ BC$ such that $ BA \equal{} BD$ and $ B$ is between $ C$ and $ D$ . Let $ I_{c}$ be center of the circle which touches $ AB$ and the extensions of $ AC$ and $ BC$ . $ CI_{c}$ intersect the circumcircle of $ ABC$ again at $ T$ . If $ \angle TDI_{c} \equal{} \frac {\angle B \plus{} \angle C}{4}$ then find $ \angle A$

The ratio of Mary's age to Alice's age is $ 3: 5$. Alice is $ 30$ years old. How old is Mary? $ \textbf{(A) } 15\qquad \textbf{(B) } 18\qquad \textbf{(C) } 20\qquad \textbf{(D) } 24\qquad \textbf{(E) } 50$

A circle of radius $a$ is revolved through $180^{\circ}$ about a line in its plane, distant $b$ from the center of the circle, where $b>a$. For what value of the ratio $\frac{b}{a}$ does the center of gravity of the solid thus generated lie on the surface of the solid?

Let $ABC$ be an isosceles triangle with $AB=4$, $BC=CA=6$. On the segment $AB$ consecutively lie points $X_{1},X_{2},X_{3},\ldots$ such that the lengths of the segments $AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{4}$. On the segment $CB$ consecutively lie points $Y_{1},Y_{2},Y_{3},\ldots$ such that the lengths of the segments $CY_{1},Y_{1}Y_{2},Y_{2}Y_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$. On the segment $AC$ consecutively lie points $Z_{1},Z_{2},Z_{3},\ldots$ such that the lengths of the segments $AZ_{1},Z_{1}Z_{2},Z_{2}Z_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$. Find all triplets of positive integers $(a,b,c)$ such that the segments $AY_{a}$, $BZ_{b}$ and $CX_{c}$ are concurrent.