Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$ and $\overline{AD}=\overline{BC}, \overline{AB}>\overline{CD}$. Let $E$ be a point such that $\overline{EC}=\overline{AC}$ and $EC \perp BC$, and $\angle ACE<90^{\circ}$. Let $\Gamma$ be a circle with center $D$ and radius $DA$, and $\Omega$ be the circumcircle of triangle $AEB$. Suppose that $\Gamma$ meets $\Omega$ again at $F(\neq A)$, and let $G$ be a point on $\Gamma$ such that $\overline{BF}=\overline{BG}$. Prove that the lines $EG, BD$ meet on $\Omega$.

Let real numbers $a,b$ such that $a\ge b\ge 0$. Prove that \[ \sqrt{a^2+b^2}+\sqrt[3]{a^3+b^3}+\sqrt[4]{a^4+b^4} \le 3a+b .\]

p1. The pattern $ABCCCDDDDABBCCCDDDDABBCCCDDDD...$ repeats to infinity. Which letter ranks in place $2533$ ? p2. Prove that if $a > 2$ and $b > 3$ then $ab + 6 > 3a + 2b$. p3. Given a rectangle $ABCD$ with size $16$ cm $\times 25$ cm, $EBFG$ is kite, and the length of $AE = 5$ cm. Determine the length of $EF$. [img]https://cdn.artofproblemsolving.com/attachments/2/e/885af838bcf1392eb02e2764f31ae83cb84b78.png[/img] p4. Consider the following series of statements. It is known that $x = 1$. Since $x = 1$ then $x^2 = 1$. So $x^2 = x$. As a result, $x^2 - 1 = x- 1$ $(x -1) (x + 1) = (x - 1) \cdot 1$ Using the rule out, we get $x + 1 = 1$ $1 + 1 = 1$ $2 = 1$ The question. a. If $2 = 1$, then every natural number must be equal to $ 1$. Prove it. b. The result of $2 = 1$ is something that is impossible. Of course there's something wrong in the argument above? Where is the fault? Why is that you think wrong? p5. To calculate $\sqrt{(1998)(1996)(1994)(1992)+16}$ . someone does it in a simple way as follows: $2000^2-2 \times 5\times 2000 + 5^2 - 5$? Is the way that person can justified? Why? p6. To attract customers, a fast food restaurant give gift coupons to everyone who buys food at the restaurant with a value of more than $25,000$ Rp.. Behind every coupon is written one of the following numbers: $9$, $12$, $42$, $57$, $69$, $21$, 15, $75$, $24$ and $81$. Successful shoppers collect coupons with the sum of the numbers behind the coupon is equal to 100 will be rewarded in the form of TV $21''$. If the restaurant owner provides as much as $10$ $21''$ TV pieces, how many should be handed over to the the customer? p7. Given is the shape of the image below. [img]https://cdn.artofproblemsolving.com/attachments/4/6/5511d3fb67c039ca83f7987a0c90c652b94107.png[/img] The centers of circles $B$, $C$, $D$, and $E$ are placed on the diameter of circle $A$ and the diameter of circle $B$ is the same as the radius of circle $A$. Circles $C$, $D$, and $E$ are equal and the pairs are tangent externally such that the sum of the lengths of the diameters of the three circles is the same with the radius of the circle $A$. What is the ratio of the circumference of the circle $A$ with the sum of the circumferences of circles $B$, $C$, $D$, and $E$? p8. It is known that $a + b + c = 0$. Prove that $a^3 + b^3 + c^3 = 3abc$.

We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?

Triangle $ABC$ lies in a regular decagon as shown in the figure. What is the ratio of the area of the triangle to the area of the entire decagon? Write the answer as a fraction of integers. [img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]

Given triangle $ABC$ and a point $P$ inside it, $\angle BAP=18^\circ$, $\angle CAP=30^\circ$, $\angle ACP=48^\circ$, and $AP=BC$. If $\angle BCP=x^\circ$, find $x$.

The vertices of the convex quadrilateral $ABCD$ and the intersection point $S$ of its diagonals are integer points in the plane. Let $P$ be the area of $ABCD$ and $P_1$ the area of triangle $ABS$. Prove that \[\sqrt{P} \ge \sqrt{P_1}+\frac{\sqrt2}2\]

Positive integer $n>2$ is called [i]good[/i] if there exist $n$ distinct points on plane($X_1, \ldots, X_n$), such that for all $1 \leq i \leq n$ vectors $X_iX_1, \ldots, X_iX_n$ can be partitioned into two groups with equal sums. Find all [i]good[/i] numbers

Let $ABC$ be a triangle in the plane with $AB = 13$, $BC = 14$, $AC = 15$. Let $M_n$ denote the smallest possible value of $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$ over all points $P$ in the plane. Find $\lim_{n \to \infty} M_n$.

Prove that there is no polyhedron whose every plane section is a triangle.

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

Given two intersecting circles with centers $O_1, O_2$. Construct the circle touching one of them externally and the second one internally such that the distance from its center to $O_1O_2$ is maximal. (M.Volchkevich)

[b]Q14.[/b] Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$.

Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that the angle bisector of its angle $\angle B$ meets the side $AC$ at a point $D$ and that $BC = BD+AD$. Determine $\angle A$.

Let $ABCD$ be a convex quadrilateral. Suppose that similar isosceles triangles $APB, BQC, CRD, DSA$ with the bases on the sides of $ABCD$ are constructed in the exterior of the quadrilateral such that $PQRS$ is a rectangle but not a square. Show that $ABCD$ is a rhombus.

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

Let $C_1,C_2$ be two circles such that the center of $C_1$ is on the circumference of $C_2$. Let $C_1,C_2$ intersect each other at points $M,N$. Let $A,B$ be two points on the circumference of $C_1$ such that $AB$ is the diameter of it. Let lines $AM,BN$ meet $C_2$ for the second time at $A',B'$, respectively. Prove that $A'B'=r_1$ where $r_1$ is the radius of $C_1$.

$P$ is a point in the interior of $\triangle ABC$, and $\angle ABP = \angle PCB = 10^\circ$. (a) If $\angle PBC = 10^\circ$ and $\angle ACP = 20^\circ$, what is the value of $\angle BAP$? (b) If $\angle PBC = 20^\circ$ and $\angle ACP = 10^\circ$, what is the value of $\angle BAP$?

Given a trapezoid with bases $AB$ and $CD$, there exists a point $E$ on $CD$ such that drawing the segments $AE$ and $BE$ partitions the trapezoid into $3$ similar isosceles triangles, each with long side twice the short side. What is the sum of all possible values of $\frac{CD}{AB}$? [i]Proposed by Adam Bertelli[/i]

Triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ and $E$ be points on the respective sides $AB$ and $AC$ so that $DE$ is perpendicular to $AO$. Show that the four points $B,D,E$ and $C$ lie on a circle.

A triangle $ABC$ and a point $P$ inside this triangle are given. Define $D, E$ and $F$ as the midpoints of $AP, BP$ and $CP$, respectively. Furthermore, let $R$ be the intersection of $AE$ and $BD, S$ the intersection of $BF$ and $CE$, and $T$ the intersection of $CD$ and $AF$. Prove that the area of hexagon $DRESFT$ is independent of the position of $P$ inside the triangle. [asy] unitsize(1 cm); pair A, B, C, D, E, F, P, R, S, T; A = (0,0); B = (5,0); C = (1.5,4); P = (2,2); D = (A + P)/2; E = (B + P)/2; F = (C + P)/2; R = extension(A,E,B,D); S = extension(B,F,C,E); T = extension(C,D,A,F); draw(A--B--C--cycle); draw(A--P); draw(B--P); draw(C--P); draw(A--F--B); draw(B--D--C); draw(C--E--A); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, N); dot("$D$", D, dir(270)); dot("$E$", E, dir(270)); dot("$F$", F, W); dot("$P$", P, dir(270)); dot("$R$", R, dir(270)); dot("$S$", S, SW); dot("$T$", T, SE); [/asy]

Let $A_1A_2 \cdots A_{10}$ be a regular decagon such that $[A_1A_4]=b$ and the length of the circumradius is $R$. What is the length of a side of the decagon? $ \textbf{a)}\ b-R \qquad\textbf{b)}\ b^2-R^2 \qquad\textbf{c)}\ R+\dfrac b2 \qquad\textbf{d)}\ b-2R \qquad\textbf{e)}\ 2b-3R $

Let $D$ be an interior point of triangle $ABC$. Lines $BD$ and $CD$ intersect sides $AC$ and $AB$ at points $E$ and $F$, respectively. Points $X$ and $Y$ are on the plane such that $BFEX$ and $CEFY$ are parallelograms. Suppose lines $EY$ and $FX$ intersect at a point $T$ inside triangle $ABC$. Prove that points $B$, $C$, $E$, and $F$ are concyclic if and only if $\angle BAD = \angle CAT$.

Find all nonconvex quadrilaterals in which the sum of the distances to the lines containing the sides is the same for any interior point. Try to generalize the result in the case of an arbitrary non-convex polygon, polyhedron.