In a convex quadrilateral $ABCD$, let $AB + CD = BC + AD$. Prove that the circle inscribed in $ABC$ is tangent to the circle inscribed in $ACD$.

In the triangle $ABC$ the incircle is tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively. The line $DE$ intersects the circumcircle at $P$ and $Q$, with $P$ in the small arc $AB$ and $Q$ in the small arc $AC$. If $P$ is the midpoint of the arc $AB$, find the angle A and the ratio $\frac{PQ}{BC}$.

Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$. Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$. Tangent form $D$ touches $\Omega$ in $E$. FInd $\angle BEC$.

From each of the vertices of a regular $n$-gon a car starts to move with constant speed along the perimeter of the $n$-gon in the same direction. Prove that if all the cars end up at a vertex $A$ at the same time, then they never again meet at any other vertex of the $n$-gon. Can they meet again at $A \ ?$

Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations \[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

Let $ABC$ be a non-isosceles triangle with altitudes $AD$, $BE$, $CF$ with orthocenter $H$. Suppose that $DF \cap HB = M$, $DE \cap HC = N$ and $T$ is the circumcenter of triangle $HBC$. Prove that $AT\perp MN$.

The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.

Let $O$ be the centre of a parallelogram $ABCD$ and $P$ be any point in the plane. Let $M, N$ be the midpoints of $AP, BP$, respectively and $Q$ be the intersection of $MC$ and $ND$. Prove that $O, P$ and $Q$ are collinear.

Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron. (S. Shosman, O. Ogievetsky)

Consider a circle with centre $O$, and $3$ points on it, $A,B$ and $C$, such that $\angle {AOB}< \angle {BOC}$. Let $D$ be the midpoint on the arc $AC$ that contains the point $B$. Consider a point $K$ on $BC$ such that $DK \perp BC$. Prove that $AB+BK=KC$.

An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

[b]p1.[/b] (a) Put the numbers $1$ to $6$ on the circle in such way that for any five consecutive numbers the sum of first three (clockwise) is larger than the sum of remaining two. (b) Can you arrange these numbers so it works both clockwise and counterclockwise. [b]p2.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p3.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p4.[/b] Two consecutive three‐digit positive integer numbers are written one after the other one. Show that the six‐digit number that is obtained is not divisible by $1001$. [b]p5.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.

Let $O$ be the circumcenter of acute $\triangle ABC$($AB<AC$), the angle bisector of $\angle BAC$ meets $BC$ at $T$ and $M$ is the midpoint of $AT$. Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$. $D,E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP, CE=CP$. If $AO$ bisects segment $DE$, prove that $AO$ is tangent to the circumcircle of $\triangle AMP$.

(a) What is the maximum area of a triangle with vertices in a given square (or on its boundary)? (b) What is the maximum volume of a tetrahedron with vertices in a given cube (or on its boundary)?

The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$.

Let $ABC$ be a triangle with bisectors $AA_1,BB_1, CC_1$ ($A_1 \in BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB = BC = CA.$

In $\vartriangle ABC$ with $AB > AC$, the tangent to the circumcircle at $A$ intersects line $BC$ at $P$. Let $Q$ be the point on $AB$ such that $AQ = AC$, and $A$ lies between $B$ and $Q$. Let $R$ be the point on ray $AP$ such that $AR = CP$. Let $X, Y$ be the midpoints of $AP, CQ$ respectively. Prove that $CR = 2XY$ .

Inside a right angle is given a point $A$. Construct an equilateral triangle, one of the vertices of which is point $A$, and two others lie on the sides of the angle (one on each side).

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

Let $ABC$ be a triangle inscribed in circle $C(O,R)$. Let $M$ ba apoint on the arc $BC$ . Let $D,E,Z$ be the feet of the perpendiculars drawn from $M$ on lines $AB,AC,BC$ respectively. Prove that $\frac{(BC)^2}{(MZ)^2} \ge 8\frac{R U_a}{(MD)\cdot(ME)}$ where $U_a$ is the altitude drawn on $BC$.

Let the incircle of $\triangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $D'$ be the diametrically opposite point of $D$ with respect to the incircle. Let lines $AD'$ and $AD$ intersect the incircle again at $X$ and $Y$, respectively. Prove that the lines $DX$, $D'Y$, and $EF$ are concurrent, i.e., the lines intersect at the same point. [i](Kritesh Dhakal, Nepal)[/i]

The perpendicular bisector of $AB$ of an acute $\Delta ABC$ intersects $BC$ and the continuation of $AC$ in points $P$ and $Q$ respectively. $M$ and $N$ are the middle points of side $AB$ and segment $PQ$ respectively. If the lines $AB$ and $CN$ intersect in point $D$, prove that $\Delta ABC$ and $\Delta DCM$ have a common orthocenter.

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.