In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$, the diagonals $AC$ and $BD$ intersect at the point $E$. Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$. Prove that the lines $AP$, $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.

Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B' + B' C' + C' A' \le DA + DB + DC$. (V.Yassinsky)

The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$ $\textbf{(A) } 20 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 24 $

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 $AB = BC = CA.$

Every vertex of a polygon has both integer coordinates; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number.

Let $ABC$ be a triangle. Consider three circles, centered at $A, B, C$, with respective radii $$\sqrt{AB \cdot AC},\sqrt{BC \cdot BA},\sqrt{CA \cdot CB}.$$ Given that there are six distinct pairwise intersections between these three circles, show that they lie on two concentric circles. [i](Two circles are concentric if they have the same center.)[/i]

Let $ABC$ be a triangle with incenter $I$. Let $K$ be the midpoint of $AI$ and $BI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N$. points $P,Q$ lie on $AM,AN$ respectively such that $\angle ABK=\angle PBC,\angle ACK=\angle QCB$. Prove that $P,Q,I$ are collinear.

In an acute triangle $ABC$ , the bisector of angle $\angle A$ intersects the circumscribed circle of the triangle $ABC$ at the point $W$. From point $W$ , a parallel is drawn to the side $AB$, which intersects this circle at the point $F \ne W$. Describe the construction of the triangle $ABC$, if given are the segments $FA$ , $FW$ and $\angle FAC$. (Andrey Mostovy)

Let $ABC$ be a triangle with incenter $I$ and let $AI$ meet $BC$ at $D$. Let $E$ be a point on the segment $AC$, such that $CD=CE$ and let $F$ be on the segment $AB$ such that $BF=BD$. Let $(CEI) \cap (DFI)=P \neq I$ and $(BFI) \cap (DEI)=Q \neq I$. Prove that $PQ \perp BC$. [i]Proposed by Leonardo Franchi, Italy[/i]

Given an acute triangle $ABC$ with circumcenter $O$. The circumcircle of $BCH$ and a circle with diameter of $AC$ intersect at $P (P \neq C)$. A point $Q$ on segment of $PC$ such that $PB = PQ$. Prove that $\angle ABC = \angle AQP$

[b]p1.[/b] BmMT is in a week, and we don’t have any problems! Let’s write $1$ on the first day, $2$ on the second day, $4$ on the third, $ 8$ on the fourth, $16$ on the fifth, $32$ on the sixth, and $64$ on the seventh. After seven days, how many problems will we have written in total? [b]p2.[/b] $100$ students are taking a ten-point exam. $50$ students scored $8$ points, $30$ students scored $7$ points, and the rest scored $9$ points. What is the average score for the exam? [b]p3.[/b] Rebecca has four pairs of shoes. Rebecca may or may not wear matching shoes. However, she will always use a left-shoe for her left foot and a right-shoe for her right foot. How many ways can Rebecca wear shoes? [b]p4.[/b] A council of $111$ mathematicians voted on whether to hold their conference in Beijing or Shanghai. The outcome of an initial vote was $70$ votes in favor of Beijing, and 41 votes in favor of Shanghai. If the vote were to be held again, what is the minimum number of mathematicians that would have to change their votes in order for Shanghai to win a majority of votes? [b]p5.[/b] What is the area of the triangle bounded by the line $20x + 16y = 160$, the $x$-axis, and the $y$-axis? [b]p6.[/b] Suppose that $3$ runners start running from the start line around a circular $800$-meter track and that their speeds are $100$, $160$, and $200$ meters per minute, respectively. How many minutes will they run before all three are next at the start line at the same time? [b]p7.[/b] Brian’s lawn is in the shape of a circle, with radius $10$ meters. Brian can throw a frisbee up to $50$ meters from where he stands. What is the area of the region (in square meters) in which the frisbee can land, if Brian can stand anywhere on his lawn? [b]p8.[/b] A seven digit number is called “bad” if exactly four of its digits are $0$ and the rest are odd. How many seven digit numbers are bad? [b]p9.[/b] Suppose you have a $3$-digit number with only even digits. What is the probability that twice that number also has only even digits? [b]p10.[/b] You have a flight on Air China from Beijing to New York. The flight will depart any time between $ 1$ p.m. and $6$ p.m., uniformly at random. Your friend, Henry, is flying American Airlines, also from Beijing to New York. Henry’s flight will depart any time between $3$ p.m. and $5$ p.m., uniformly at random. What is the probability that Henry’s flight departs before your flight? [b]p11.[/b] In the figure below, three semicircles are drawn outside the given right triangle. Given the areas $A_1 = 17$ and $A_2 = 14$, find the area $A_3$. [img]https://cdn.artofproblemsolving.com/attachments/4/4/28393acb3eba83a5a489e14b30a3e84ffa60fb.png[/img] [b]p12.[/b] Consider a circle of radius $ 1$ drawn tangent to the positive $x$ and $y$ axes. Now consider another smaller circle tangent to that circle and also tangent to the positive $x$ and $y$ axes. Find the radius of the smaller circle. [img]https://cdn.artofproblemsolving.com/attachments/7/4/99b613d6d570db7ee0b969f57103d352118112.png[/img] [b]p13.[/b] The following expression is an integer. Find this integer: $\frac{\sqrt{20 + 16\frac{\sqrt{20+ 16\frac{20 + 16...}{2}}}{2}}}{2}$ [b]p14.[/b] Let $2016 = a_1 \times a_2 \times ... \times a_n$ for some positive integers $a_1, a_2, ... , a_n$. Compute the smallest possible value of $a_1 + a_2 + ...+ a_n$. [b]p15.[/b] The tetranacci numbers are defined by the recurrence $T_n = T_{n-1} + T_{n-2} + T_{n-3} + T_{n-4}$ and $T_0 = T_1 = T_2 = 0$ and $T_3 = 1$. Given that $T_9 = 29$ and $T_{14} = 773$, calculate $T_{15}$. [b]p16.[/b] Find the number of zeros at the end of $(2016!)^{2016}$. Your answer should be an integer, not its prime factorization. [b]p17.[/b] A DJ has $7$ songs named $1, 2, 3, 4, 5, 6$, and $7$. He decides that no two even-numbered songs can be played one after the other. In how many different orders can the DJ play the $7$ songs? [b]p18.[/b] Given a cube, how many distinct ways are there (using $6$ colors) to color each face a distinct color? Colorings are distinct if they cannot be transformed into one another by a sequence of rotations. [b]p19. [/b]Suppose you have a triangle with side lengths $3, 4$, and $5$. For each of the triangle’s sides, draw a square on its outside. Connect the adjacent vertices in order, forming $3$ new triangles (as in the diagram). What is the area of this convex region? [img]https://cdn.artofproblemsolving.com/attachments/4/c/ac4dfb91cd055badc07caface93761453049fa.png[/img] [b]p20.[/b] Find $x$ such that $\sqrt{c +\sqrt{c - x}} = x$ when $c = 4$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A', B', C', D', E'$ and $F'$ be the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EF$ and $FA$ of an arbitrary convex hexagon $ABCDEF$ respectively. Express the area of $ABCDEF$ in terms of the areas of the triangles $ABC$, $BCD'$, $CDS'$, $DEF'$, $EFA'$ and $FAB'$. (A Lopshi tz, NB Vassiliev)

In triangle $ABC$, let $D$ be on side $BC$. The line through $D$ parallel to $AB,AC$ meet $AC,AB$ at $E,F$, respectively. (a) Show that if $D$ varies on line $BC$, the circumcircle of $AEF$ passes through a fixed point $T$. (b) Show that if $D$ lies on line $AT$, then the circumcircle of $AEF$ is tangent to the circumcircle of $BTC$.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

In a rectangle triangle, let $I$ be its incenter and $G$ its geocenter. If $IG$ is parallel to one of the catheti and measures $10 cm$, find the lengths of the two catheti of the triangle.

Let $ABC$ be a scalene triangle, and let $D$ be a point on side $BC$ satisfying $\angle BAD=\angle DAC$. Suppose that $X$ and $Y$ are points inside $ABC$ such that triangles $ABX$ and $ACY$ are similar and quadrilaterals $ACDX$ and $ABDY$ are cyclic. Let lines $BX$ and $CY$ meet at $S$ and lines $BY$ and $CX$ meet at $T$. Prove that lines $DS$ and $AT$ are parallel. [i]Michael Ren[/i]

Areas of four surfaces of a tetrahedron are $S_1,S_2,S_3,S_4$. And the largest one of them is $S$. $\lambda=\frac{S_1+S_2+S_3+S_4}{S}$, then $\lambda$ always satisfies $\text{(A)}2<\lambda\leq4\qquad\text{(B)}3<\lambda<4\qquad\text{(C)}2.5<\lambda\leq4.5\qquad\text{(D)}3.5<\lambda<5.5$

In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$,$\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$. Then find the angle $\angle{DBA}$

Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$

In $\triangle ABC$, $AD$ is the bisector of $\angle A$, and $E$, $F$ are the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. $H$ is the intersection of $BE$ and $CF$, and $G$, $I$ are the feet of the perpendiculars from $D$ to $BE$ and $CF$, respectively. Prove that both $AFEH$ and $AEIH$ are cyclic quadrilaterals.

Let $C$ and $D$ be two points, not diametrically opposite, on a circle $C_1$ with center $M$. Let $H$ be a point on minor arc $CD$. The tangent to $C_1$ at $H$ intersects the circumcircle of $CMD$ at points $A$ and $B$. Prove that $CD$ bisects $MH$ iff $\angle AMB = \frac{\pi}{2}$.

[b]p1.[/b] $ x,y,z$ are required to be non-negative whole numbers, find all solutions to the pair of equations $$x+y+z=40$$ $$2x + 4y + 17z = 301.$$ [b]p2.[/b] Let $P$ be a point lying between the sides of an acute angle whose vertex is $O$. Let $A,B$ be the intersections of a line passing through $P$ with the sides of the angle. Prove that the triangle $AOB$ has minimum area when $P$ bisects the line segment $AB$. [b]p3.[/b] Find all values of $x$ for which $|3x-2|+|3x+1|=3$. [b]p4.[/b] Prove that $x^2+y^2+z^2$ cannot be factored in the form $$(ax + by + cz) (dx + ey + fz),$$ $a, b, c, d, e, f$ real. [b]p5.[/b] Let $f(x)$ be a continuous function for all real values of $x$ such that $f(a)\le f(b)$ whenever $a\le b$. Prove that, for every real number $r$, the equation $$x + f(x) = r$$ has exactly one solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

In an acute triangle $ABC$, point $D$ is on the segment $AC$ such that $\overline{AD}=\overline{BC}$ and $\overline{AC}^2-\overline{AD}^2=\overline{AC}\cdot\overline{AD}$. The line that is parallel to the bisector of $\angle{ACB}$ and passes the point $D$ meets the segment $AB$ at point $E$. Prove, if $\overline{AE}=\overline{CD}$, $\angle{ADB}=3\angle{BAC}$.

Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If T denotes the sum of the numbers on all the faces, find the possible values of T.