Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively. a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$. b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?

Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.

$D$ is an arbitrary point lying on side $BC$ of $\triangle{ABC}$. Circle $\omega_1$ is tangent to segments $AD$ , $BD$ and the circumcircle of $\triangle{ABC}$ and circle $\omega_2$ is tangent to segments $AD$ , $CD$ and the circumcircle of $\triangle{ABC}$. Let $X$ and $Y$ be the intersection points of $\omega_1$ and $\omega_2$ with $BC$ respectively and take $M$ as the midpoint of $XY$. Let $T$ be the midpoint of arc $BC$ which does not contain $A$. If $I$ is the incenter of $\triangle{ABC}$, prove that $TM$ goes through the midpoint of $ID$.

$ABCD$ is a circumscribed and inscribed quadrilateral. $O$ is the circumcenter of the quadrilateral. $E,F$ and $S$ are the intersections of $AB,CD$ , $AD,BC$ and $AC,BD$ respectively. $E'$ and $F'$ are points on $AD$ and $AB$ such that $A\hat{E}E'=E'\hat{E}D$ and $A\hat{F}F'=F'\hat{F}B$. $X$ and $Y$ are points on $OE'$ and $OF'$ such that $\frac{XA}{XD}=\frac{EA}{ED}$ and $\frac{YA}{YB}=\frac{FA}{FB}$. $M$ is the midpoint of arc $BD$ of $(O)$ which contains $A$. Prove that the circumcircles of triangles $OXY$ and $OAM$ are coaxal with the circle with diameter $OS$.

Let $ABC$ be triangle with circumcircle $(O)$ of fixed $BC$, $AB \ne AC$ and $BC$ not a diameter. Let $I$ be the incenter of the triangle $ABC$ and $D = AI \cap BC, E = BI \cap CA, F = CI \cap AB$. The circle passing through $D$ and tangent to $OA$ cuts for second time $(O)$ at $G$ ($G \ne A$). $GE, GF$ cut $(O)$ also at $M, N$ respectively. a) Let $H = BM \cap CN$. Prove that $AH$ goes through a fixed point. b) Suppose $BE, CF$ cut $(O)$ also at $L, K$ respectively and $AH \cap KL = P$. On $EF$ take $Q$ for $QP = QI$. Let $J$ be a point of the circimcircle of triangle $IBC$ so that $IJ \perp IQ$. Prove that the midpoint of $IJ$ belongs to a fixed circle.

[size=130][b]Higher Secondary: 2011[/b] [/size] Time: 4 Hours [b]Problem 1:[/b] Prove that for any non-negative integer $n$ the numbers $1, 2, 3, ..., 4n$ can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=709 [b]Problem 2:[/b] In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $90$. How many players were there in the tournament? http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=708 [b]Problem 3:[/b] $E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=683 [b]Problem 4:[/b] Which one is larger 2011! or, $(1006)^{2011}$? Justify your answer. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=707 [b]Problem 5:[/b] In a scalene triangle $ABC$ with $\angle A = 90^{\circ}$, the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R$. The lines $RS$ and $BC$ intersect at $N$ while the lines $AM$ and $SR$ intersect at $U$. Prove that the triangle $UMN$ is isosceles. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=706 [b]Problem 6:[/b] $p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=693 [b]Problem 7:[/b] Consider a group of $n > 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ [b]friendship/enmity[/b]. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of $n$. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=694 [b]Problem 8:[/b] $ABC$ is a right angled triangle with $\angle A = 90^{\circ}$ and $D$ be the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\dfrac{2}{\sqrt{15}}$ http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=704 [b]Problem 9:[/b] The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=703 [b]Problem 10:[/b] Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=702 The problems of the Junior categories are available in [url=http://matholympiad.org.bd/forum/]BdMO Online forum[/url]: http://matholympiad.org.bd/forum/viewtopic.php?f=25&t=678

Let $ABCD$ be a circumscribed quadrilateral, assume $ABCD$ is not a kite. Denote the circumcenters of triangle $ABC,BCD,CDA,DAB$ by $O_D,O_A,O_B,O_C$ respectively. a. Prove that $O_AO_BO_CO_D$ is circumscribed. b. Let the angle bisector of $\angle BAD$ intersect the angle bisector of $\angle O_BO_AO_D$ in $X$. Similarly define the points $Y,Z,W$. Denote the incenters of $ABCD, O_AO_BO_CO_D$ by $I,J$ respectively. Express the angles $\angle ZYJ,\angle XYI$ in terms of angles of quadrilateral $ABCD$.

In an acute triangle $AB$ the heights $ BE$ and $CF$ intersect at the orthocenter $H$, and $M$ is the midpoint of $BC$. The line $EF$ intersects the lines $MH$ and $BC$ at the points $P$ and $T$ , respectively. $AP$ intersects the cirumcscribed circle of $\vartriangle ABC$ for second time at the point $Q$ . Prove that $\angle AQT= 90^o$. (Fedir Yudin)

A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$

In triangle $ABC$, let $A'$ is the symmetrical of $A$ with respect to the circumcenter $O$ of $ABC$. Prove that: [b]a)[/b] The sum of the squares of the tangents segments drawn from $A$ and $A'$ to the incircle of $ABC$ equals $$4R^2-4Rr-2r^2$$ where $R$ and $r$ are the radii of the circumscribed and inscribed circles of $ABC$ respectively. [b]b)[/b] The circle with center $A'$ and radius $A'I$ intersects the circumcircle of $ABC$ in a point $L$ such that $$AL=\sqrt{ AB.AC}$$ where $I$ is the centre of the inscribed circle of $ABC$.

Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property: For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.

Given two circles $k_{1}$ and $k_{2}$ with centers $O_{1}$ and $O_{2}$ that intersect at the points $A$ and $B$.Passes through A two lines that intersect the circle $k_{1}$ at the points $N_{1}$and $M_{1}$, and the circle $k_{2}$ at the points $N_{2}$ and $M_{2}$ (points $A, N_{1},M_{1}$ in colinear). Denote the midpoints of the segments $N_{1}N_{2}$ and $M_{1}M_{2]}$ , through $N$ and $M$.Prove that: $a)$ Points $M,N,A$ and $B$ lie on a circle $b)$The center of the circle passing through $M,N,A$ and $B$ lies in the middle of the segment $O_{1}O_{2}$

Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$. [i]Cezar Lupu & Vlad Matei[/i]

Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon. [i]Jordan Tabov[/i]

In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

Let $ABC$ be a triangle and let $\Gamma$ be its circumcircle. Let $D$ be a point on $AB$ such that $CD$ is parallel to the line tangent to $\Gamma$ at $A$. Let $E$ be the intersection of $CD$ with $\Gamma$ distinct from $C$, and $F$ the intersection of $BC$ with the circumcircle of $\bigtriangleup ADC$ distinct from $C$. Finally, let $G$ be the intersection of the line $AB$ and the internal bisector of $\angle DCF$. Show that $E,\ G,\ F$ and $C$ lie on the same circle.

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects the segments $AB$ and $AC$ at the interior points $D$ and $E$, respectively. The lines $BE$ and $CD$ intersects at $F$. Let $G$ be a point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$ and let $H$ be a point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T\neq A$, independent of the choice of $\omega$, such that the circumcircle of triangle $AGH$ passes through $T$.

The incircle of triangle $ABC$ meet the sides $AB, AC$ and $BC$ in $M,N$ and $P$, respectively. Prove that the orthocenter of triangle $MNP,$ the incenter and the circumcenter of triangle $ABC$ are collinear. [asy] import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffwwww = rgb(1,0.4,0.4); pen xdxdff = rgb(0.49,0.49,1); draw((8,17.58)--(2.84,9.26)--(20.44,9.21)--cycle); draw((8,17.58)--(2.84,9.26),ttttff+linewidth(2pt)); draw((2.84,9.26)--(20.44,9.21),ttttff+linewidth(2pt)); draw((20.44,9.21)--(8,17.58),ttttff+linewidth(2pt)); draw(circle((9.04,12.66),3.43),blue+linewidth(1.2pt)+linetype("8pt 8pt")); draw((6.04,14.42)--(8.94,9.24),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(11.12,15.48),ffwwww+linewidth(1.2pt)); draw((11.12,15.48)--(6.04,14.42),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(7.81,14.79)); draw((11.12,15.48)--(6.95,12.79)); draw((6.04,14.42)--(10.12,12.6)); dot((8,17.58),ds); label("$A$", (8.11,18.05),NE*lsf); dot((2.84,9.26),ds); label("$B$", (2.11,8.85), NE*lsf); dot((20.44,9.21),ds); label("$C$", (20.56,8.52), NE*lsf); dot((9.04,12.66),ds); label("$O$", (8.94,12.13), NE*lsf); dot((6.04,14.42),ds); label("$M$", (5.32,14.52), NE*lsf); dot((11.12,15.48),ds); label("$N$", (11.4,15.9), NE*lsf); dot((8.94,9.24),ds); label("$P$", (8.91,8.58), NE*lsf); dot((7.81,14.79),ds); label("$D$", (7.81,15.14),NE*lsf); dot((6.95,12.79),ds); label("$F$", (6.64,12.07),NE*lsf); dot((10.12,12.6),ds); label("$G$", (10.41,12.35),NE*lsf); dot((8.07,13.52),ds); label("$H$", (8.11,13.88),NE*lsf); clip((-0.68,-0.96)--(-0.68,25.47)--(30.71,25.47)--(30.71,-0.96)--cycle); [/asy]

Let $I$ be the incenter and $AD$ be a diameter of the circumcircle of a triangle $ABC.$ If the point $E$ on the ray $BA$ and the point $F$ on the ray $CA$ satisfy the condition \[BE=CF=\frac{AB+BC+CA}{2}\] show that the lines $EF$ and $DI$ are perpendicular.

On the side $AB$ of a triangle $ABC$, two points $X, Y$ are chosen so that $AX = BY$. Lines $CX$ and $CY$ meet the circumcircle of the triangle, for the second time, at points $U$ and $V$. Prove that all lines $UV$ (for all $X, Y$, given $A, B, C$) have a common point.

Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.

Let ABC be a triangle. The external and internal angle bisectors of ∠CAB intersect side BC at D and E, respectively. Let F be a point on the segment BC. The circumcircle of triangle ADF intersects AB and AC at I and J, respectively. Let N be the mid-point of IJ and H the foot of E on DN. Prove that E is the incenter of triangle AHF, or the center of the excircle. [i]Proposed by Steve Dinh[/i]

Let $M, N$ be points inside the angle $\angle BAC$ usch that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.

In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.