In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find the acute angle between the diagonals $AC$ and $BD$.

Let be a convex quadrilateral $ ABCD $ and the points $ M,N,P,Q $ such that $ MAB\sim NBC\sim PCD\sim QDA. $ [b]a)[/b] Prove that $ ABCD $ is a parallelogram if and only if $ MNPQ $ is a parallelogram. [b]b)[/b] Show that if the diagonals of $ MNPQ $ are congruent and perpendicular, then the diagonals of $ ABCD $ are congruent and perpendicular, or $ MAB $ is a right isosceles triangle. [i]Nelu Chichirim[/i]

We have $4$ sticks. It is known, that for every $3$ sticks we can build a triangle with the same area. Is it true, that sticks have the same length?

Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.

Given a triangular pyramid in which all the plane angles at one of the vertices are right. It is known that there is a point in space located at a distance of $3$ from the indicated vertex and at distances $\sqrt5, \sqrt6, \sqrt7$ from three other vertices. Find the radius of the sphere circumscribed around this pyramid. (The circumscribed sphere for a pyramid is the sphere containing all its vertices.)

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

Given are $n$ pairwise intersecting convex $k$-gons on the plane. Any of them can be transferred to any other by a homothety with a positive coefficient. Prove that there is a point in a plane belonging to at least $1 +\frac{n-1}{2k}$ of these $k$-gons.

A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$

Two circles $\Gamma_1, \Gamma_2$ with radii $r_i, r_2$, respectively, touch internally at the point $P$. A tangent parallel to the diameter through $P$ touches $ \Gamma_1$ at $R$ and intersects $\Gamma_2$ at $M$ and $N$. Prove that $PR$ bisects $\angle MPN$.

Let be given a semi-sphere $\Sigma$ whose base-circle lies on plane $p$. A variable plane $Q$, parallel to a fixed plane non-perpendicular to $P$, cuts $\Sigma$ at a circle $C$. We denote by $C'$ the orthogonal projection of $C$ onto $P$. Find the position of $Q$ for which the cylinder with bases $C$ and $C'$ has the maximum volume.

Let $\mathcal{S}$ be the set of all points in the plane. Find all functions $f : \mathcal{S} \rightarrow \mathbb{R}$ such that for all nondegenerate triangles $ABC$ with orthocenter $H$, if $f(A) \leq f(B) \leq f(C)$, then $$f(A) + f(C) = f(B) + f(H).$$

Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most $2\sqrt{2}$ times the sum of their radii. (A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.) [color=#45818E]Note.[/color] Weaker results with $2\sqrt{2}$ replaced by some other $c$ may be awarded points depending on the value of $c>2\sqrt{2}$ [i]Proposed by Morteza Saghafian[/i]

The diagonals of quadrilateral $ABCD$ intersect in point $E$. Given that $|AB| =|CE|$, $|BE| = |AD|$, and $\angle AED = \angle BAD$, determine the ratio $|BC|:|AD|$.

In an acute triangle $ABC$, a point $P$ is taken on the $A$-altitude. Lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at points $D$ and $E$, respectively. Tangents drawn from points $D$ and $E$ to the circumcircle of triangle $BPC$ are tangent to it at points $K$ and $L$, respectively, which are in the interior of triangle $ABC$. Line $KD$ intersects the circumcircle of triangle $AKC$ at point $M$ for the second time, and line $LE$ intersects the circumcircle of triangle $ALB$ at point $N$ for the second time. Prove that\[ \frac{KD}{MD}=\frac{LE}{NE} \iff \text{Point P is the orthocenter of triangle ABC}\]

Let $P_0$, $P_1$, $P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle P_{i-1} P_{i-2} P_{i-3}$. (1) Prove that the points $P_1, P_5, P_9, P_{13},\cdots$ are collinear. (2) Let $x$ be the distance from $P_1$ to $P_{1001}$, and let $y$ be the distance from $P_{1001}$ to $P_{2001}$. Determine all values of $t$ for which $\sqrt[500]{ \frac xy}$ is an integer.

Let $O$ be the center of the circle of the triangle $ABC$ with $AB \ne AC$. Furthermore, let $S$ and $T$ be points on the rays $AB$ and $AC$, such that $\angle ASO = \angle ACO$ and $\angle ATO = \angle ABO$. Show that $ST$ bisects the segment $BC$.

$I$ is incenter of $\triangle{ABC}$. The incircle touches $BC,CA,AB$ at $D,E,F$, respectively . Let $M,N,K=BI,CI,DI \cap EF$ respectively and $BN\cap CM=P,AK\cap BC=G$. Point $Q$ is intersection of the perpendicular line to $PG$ through $I$ and the perpendicular line to $PB$ through $P$. Prove that $BI$ bisect segment $PQ$.

[b]p1.[/b] Let $X,Y ,Z$ be nonzero real numbers such that the quadratic function $X t^2 - Y t + Z = 0$ has the unique root $t = Y$ . Find $X$. [b]p2.[/b] Let $ABCD$ be a kite with $AB = BC = 1$ and $CD = AD =\sqrt2$. Given that $BD =\sqrt5$, find $AC$. [b]p3.[/b] Find the number of integers $n$ such that $n -2016$ divides $n^2 -2016$. An integer $a$ divides an integer $b$ if there exists a unique integer $k$ such that $ak = b$. [b]p4.[/b] The points $A(-16, 256)$ and $B(20, 400)$ lie on the parabola $y = x^2$ . There exists a point $C(a,a^2)$ on the parabola $y = x^2$ such that there exists a point $D$ on the parabola $y = -x^2$ so that $ACBD$ is a parallelogram. Find $a$. [b]p5.[/b] Figure $F_0$ is a unit square. To create figure $F_1$, divide each side of the square into equal fifths and add two new squares with sidelength $\frac15$ to each side, with one of their sides on one of the sides of the larger square. To create figure $F_{k+1}$ from $F_k$ , repeat this same process for each open side of the smallest squares created in $F_n$. Let $A_n$ be the area of $F_n$. Find $\lim_{n\to \infty} A_n$. [img]https://cdn.artofproblemsolving.com/attachments/8/9/85b764acba2a548ecc61e9ffc29aacf24b4647.png[/img] [b]p6.[/b] For a prime $p$, let $S_p$ be the set of nonnegative integers $n$ less than $p$ for which there exists a nonnegative integer $k$ such that $2016^k -n$ is divisible by $p$. Find the sum of all $p$ for which $p$ does not divide the sum of the elements of $S_p$ . [b]p7. [/b] Trapezoid $ABCD$ has $AB \parallel CD$ and $AD = AB = BC$. Unit circles $\gamma$ and $\omega$ are inscribed in the trapezoid such that circle $\gamma$ is tangent to $CD$, $AB$, and $AD$, and circle $\omega$ is tangent to $CD$, $AB$, and $BC$. If circles $\gamma$ and $\omega$ are externally tangent to each other, find $AB$. [b]p8.[/b] Let $x, y, z$ be real numbers such that $(x+y)^2+(y+z)^2+(z+x)^2 = 1$. Over all triples $(x, y, z)$, find the maximum possible value of $y -z$. [b]p9.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, and $CA = 15$. Let $P$ be a point on segment $BC$ such that $\frac{BP}{CP} = 3$, and let $I_1$ and $I_2$ be the incenters of triangles $\vartriangle ABP$ and $\vartriangle ACP$. Suppose that the circumcircle of $\vartriangle I_1PI_2$ intersects segment $AP$ for a second time at a point $X \ne P$. Find the length of segment $AX$. [b]p10.[/b] For $1 \le i \le 9$, let Ai be the answer to problem i from this section. Let $(i_1,i_2,... ,i_9)$ be a permutation of $(1, 2,... , 9)$ such that $A_{i_1} < A_{i_2} < ... < A_{i_9}$. For each $i_j$ , put the number $i_j$ in the box which is in the $j$th row from the top and the $j$th column from the left of the $9\times 9$ grid in the bonus section of the answer sheet. Then, fill in the rest of the squares with digits $1, 2,... , 9$ such that $\bullet$ each bolded $ 3\times 3$ grid contains exactly one of each digit from $ 1$ to $9$, $\bullet$ each row of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$, and $\bullet$ each column of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$. PS. You had better use hide for answers.

The incircle of triangle $\triangle ABC$ touches $AC$ and $BC$ at $E$ and $D$ respectively. The excircle corresponding to $A$ touches the extensions of $BC$ at $A_1$, $CA$ at $B_1$, and $AB$ at $C_1$. Let $DE \cap A_1B_1 = L$. Prove that $L$ belongs to the circumcircle of triangle $\triangle A_1B_1C_1$.

Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.

Let $ ABCD $ be a convex quadrilateral such that $ AD = DC, AC = AB $ and $ \angle ADC = \angle CAB $. If $ M $ and $ N $ are midpoints of the $ AD $ and $ AB $ sides, prove that the $ MNC $ triangle is isosceles.

Chitoge is painting a cube; she can paint each face either black or white, but she wants no vertex of the cube to be touching three faces of the same color. In how many ways can Chitoge paint the cube? Two paintings of a cube are considered to be the same if you can rotate one cube so that it looks like the other cube.

The vertices of a right-angled triangle are on a circle of radius $R$ and the sides of the triangle are tangent to another circle of radius $r$ (this is the circle that is inside triangle). If the lengths of the sides about the right angles are 16 and 30, determine the value of $R+r$.

In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.

A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have? $\textbf{(A) }36\qquad \textbf{(B) }60\qquad \textbf{(C) }72\qquad \textbf{(D) }84\qquad \textbf{(E) }108\qquad$