Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$? $ \textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }11 \qquad \textbf{(D) }12 \qquad \textbf{(E) }13 \qquad $

Let $ABC$ be an acute and scalene triangle with $AB<AC$, inscribed in a circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ intersects side $BC$ at point $E$ and circle $c$ at point $F$. $EF$ intersects for the second time circle $c$ at point $D$ and side $AC$ at point $M$. $AD$ intersects $BC$ at point $K$. Circumcircle of triangle $BKD$ intersects $AB$ at point $L$ . Prove that points $K,L,M$ lie on a line parallel to $BF$.

In a scalene triangle $ABC$, $D$ is the foot of the altitude through $A$, $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$. Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$, $Y=BE\cap CF$, $Z=CF \cap AD$. If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$, $OYZ$, $OZX$ must be equilateral.

Let in triangle $ABC$ incircle touches sides $AB,BC,CA$ at $C_1,A_1,B_1$ respectively. Let $\frac {2}{CA_1}=\frac {1}{BC_1}+\frac {1}{AC_1}$ .Prove that if $X$ is intersection of incircle and $CC_1$ then $3CX=CC_1$

p1. Find an integer that has the following properties: a) Every two adjacent digits in the number are prime. b) All prime numbers referred to in item (a) above are different. p2. Determine all integers up to $\sqrt{50+\sqrt{n}}+\sqrt{50-\sqrt{n}}$ p3. The following figure shows the path to form a series of letters and numbers “OSN2015”. Determine as many different paths as possible to form the series of letters and numbers by following the arrows. [img]https://cdn.artofproblemsolving.com/attachments/6/b/490a751457871184a506c2966f8355f20cebbd.png[/img] p4. Given an acute triangle $ABC$ with $L$ as the circumcircle. From point $A$, a perpendicular line is drawn on the line segment $BC$ so that it intersects the circle $L$ at point $X$. In a similar way, a perpendicular line is made from point $B$ and point $C$ so that it intersects the circle $L$, at point $Y$ and point $Z$, respectively. Is arc length $AY$ = arc length $AZ$ ? p5. The students of class VII.3 were divided into five groups: $A, B, C, D$ and $E$. Each group conducted five science experiments for five weeks. Each week each group performs an experiment that is different from the experiments conducted by other groups. Determine at least two possible trial schedules in week five, based on the following information: $\bullet$ In the first week, group$ D$ did experiment $4$. $\bullet$ In the second week, group $C$ did the experiment $5$. $\bullet$ In the third week, group $E$ did the experiment $5$. $\bullet$ In the fourth week, group $A$ did experiment $4$ and group $D$ did experiment $2$.

Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle CDP$ intersect at $X$. The area of the locus of $X$ as $l$ varies can be expressed in the form $\tfrac{p}{q}\pi$ for positive coprime integers $p$ and $q$. What is $p + q$?

For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$

A large rectangle is tiled by some $1\times1$ tiles. In the center there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is five tiles wide. That red border is surrounded by a white border which is five tiles wide. Finally, the white border is surrounded by a red border which is five tiles wide. The resulting pattern is pictured below. In all, $2900$ red tiles are used to tile the large rectangle. Find the perimeter of the large rectangle. [asy] import graph; size(5cm); fill((-5,-5)--(0,-5)--(0,35)--(-5,35)--cycle^^(50,-5)--(55,-5)--(55,35)--(50,35)--cycle,red); fill((0,30)--(0,35)--(50,35)--(50,30)--cycle^^(0,-5)--(0,0)--(50,0)--(50,-5)--cycle,red); fill((-15,-15)--(-10,-15)--(-10,45)--(-15,45)--cycle^^(60,-15)--(65,-15)--(65,45)--(60,45)--cycle,red); fill((-10,40)--(-10,45)--(60,45)--(60,40)--cycle^^(-10,-15)--(-10,-10)--(60,-10)--(60,-15)--cycle,red); fill((-10,-10)--(-5,-10)--(-5,40)--(-10,40)--cycle^^(55,-10)--(60,-10)--(60,40)--(55,40)--cycle,white); fill((-5,35)--(-5,40)--(55,40)--(55,35)--cycle^^(-5,-10)--(-5,-5)--(55,-5)--(55,-10)--cycle,white); for(int i=0;i<16;++i){ draw((-i,-i)--(50+i,-i)--(50+i,30+i)--(-i,30+i)--cycle,linewidth(.5)); } [/asy]

$ABCD$ - convex quadrilateral. $P$ is such point on $AC$ and inside $\triangle ABD$, that $$\angle ACD+\angle BDP = \angle ACB+ \angle DBP = 90-\angle BAD$$. Prove that $\angle BAD+ \angle BCD =90$ or $\angle BDA + \angle CAB = 90$

Among the triangles that have a side of length $5$ m and the angle opposite of $30^o$, determine the one with maximum area, calculating the value of the other two angles and area of triangle.

Can the regular octahedron be inscribed into regular dodecahedron in such way that all vertices of octahedron be the vertices of dodecahedron? (B.Frenkin)

Let $(C)$ and $(L)$ be a circle and a line. $P_{1},\dots,P_{2n+1}$ are odd number of points on $(L)$. $A_{1}$ is an arbitrary point on $(C)$. $A_{k+1}$ is the intersection point of $A_{k}P_{k}$ and $(C)$ ($1\leq k\leq 2n+1$). Prove that $A_{1}A_{2n+2}$ passes through a constant point while $A_{1}$ varies on $(C)$.

Given a parallelogram $ABCD$. A circle passing through $A$ meets the line segments $AB, AC$ and $AD$ at inner points $M,K,N$, respectively. Prove that \[|AB|\cdot |AM | + |AD|\cdot |AN|=|AK|\cdot |AC|\]

Let $ABC$ be an equilateral triangle of unit side length and suppose $D$ is a point on segment $\overline{BC}$ such that $DB<DC.$ Let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC},$ respectively. Suppose $X$ and $Y$ are the intersections of lines $AB$ and $ND,$ and lines $AC$ and $MD,$ respectively. Given that $XY=4,$ what is the value of $\frac{DB}{DC}?$ [i]Proposed by Kyle Lee[/i]

Let $ABCD$ be a convex circumscribed quadrilateral such that $\angle ABC+\angle ADC<180^{\circ}$ and $\angle ABD+\angle ACB=\angle ACD+\angle ADB$. Prove that one of the diagonals of quadrilateral $ABCD$ passes through the other diagonals midpoint.

2-There is given a trapezoid $ ABCD$.We have the following properties:$ AD\parallel{}BC,DA \equal{} DB \equal{} DC,\angle BCD \equal{} 72^\circ$. A point $ K$ is taken on $ BD$ such that $ AD \equal{} AK,K \neq D$.Let $ M$ be the midpoint of $ CD$.$ AM$ intersects $ BD$ at $ N$.PROVE $ BK \equal{} ND$.

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

In the triangle $ABC$, $\angle ABC = \angle ACB = 78^o$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^o$, $\angle CBE = 51^o$. Find the measure of angle $\angle BED$.

Given a triangle $ABC$, let $P$ be a point inside the triangle such that $| BP | > | AP |, | BP | > | CP |$. Show that $\angle ABC <90^o$

$1956$ points are chosen in a cube with edge $13$. Is it possible to fit inside the cube a cube with edge $1$ that would not contain any of the selected points?

There are a number of arcs on the edge of a circular disk. Each pair of arcs has the least one point in common. Show that on the circle you can choose two diametrical opposites points such that each arc contains at least one of these two points.

Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?

Let $ABC$ be a triangle with $\angle BAC =60^\circ$. $A'$ is the symmetric point of $A$ wrt $\overline{BC}$. $D$ is the point in $\overline{AC}$ such that $\overline{AB}=\overline{AD}$. $H$ is the orthocenter of triangle $ABC$. $l$ is the external angle bisector of $\angle BAC$. $\{M\}=\overline{A'D}\cap l$,$\{N\}=\overline{CH} \cap l$. Show that $\overline{AM}=\overline{AN}$.

Let ABCD be a quadrilateral; X and Y be the midpoints of AC and BD respectively and lines through X and Y respectively parallel to BD, AC meet in O. Let P,Q,R,S be the midpoints of AB, BC, CD, DA respectively. Prove that (A) APOS and APXS have the same area (B) APOS, BQOP, CROQ, DSOR have the same area.

There is a sphere $ K $ in space and points $ A, B $ outside the sphere such that the segment $ AB $ intersects the interior of the sphere. Prove that the set of points $ P $ for which the segments $ AP $ and $ BP $ are tangent to the sphere $ K $ is contained in a certain plane.