Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 75^o$ and $AB = 2$. The points $P$ and $Q$ on the sides $AC$ and $BC$ respectively are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$ . Calculate the measurement of the segment $QA $.

There is a circle $k$ in a plane with center $M$ and radius $r$. The following illustration, through which every point $P \ne M$., is called a “reflection on the circle $k$” from $\varepsilon$ a point $P'$ from $\varepsilon$ is assigned: (1) $P'$ lies on the ray emanating from$ M$ and passing through $P$. (2) It is $MP \cdot MP' = r^2$. a) Construct the mirror point $ P'$ for any given point $P \ne M$ inside $k$. b) Let another circle $k_1$ be given arbitrarily, but such that $M$ lies outside $k_1$.Construct $k'_1$ , i.e. the set of all mirror points $P'$ of the points $P$ of $k_1$.

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

In convex quadrilateral $AEBC$, $\angle BEA = \angle CAE = 90^{\circ}$ and $AB = 15$, $BC = 14$ and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to $\overline{AB}$. If ray $CD$ meets $\overline{AE}$ at $F$, compute $AE \cdot AF$. [i]Proposed by David Stoner[/i]

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

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]

Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$. Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$. (a) Prove that $L$ is nonempty. (b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$. [i]Proposed by David Yang[/i]

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

Let be a triangle $ ABC $ that's not equilateral, nor right-angled. Let $ A',B',C' $ be the feet of the heights of $ A,B,C, $ respectively. Prove that the Euler's lines of the triangles $ AB'C',BC'A',CA'B' $ meet at one point on the Euler's circle of $ ABC. $

Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be feet of the altitudes from $B$ and $C$ respectively. Let $M$ be the midpoint of segment $AH$ and $F$ be the intersection point of $AH$ and $DE$. Furthermore, let $P$ and $Q$ be the points inside triangle $ADE$ so that $P$ is an intersection of $CM$ and the circumcircle of $DFH$, and $Q$ is an intersection of $BM$ and the circumcircle of $EFH$. Prove that the intersection of lines $DQ$ and $EP$ lies on segment $AH$.

A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $a/b$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals: $\textbf{(A) }\sqrt a:\sqrt b\qquad \textbf{(B) }\sqrt a:\sqrt{a-b}\qquad \textbf{(C) }\sqrt b:\sqrt{a-b}\qquad$ $\textbf{(D) }a:\sqrt{a-b}\qquad \textbf{(E) }b:\sqrt{a-b}$

Let $O$ and $P$ be fixed points on a plane, and let $ABCD$ be any parallelogram with center $O$. Let $M$ and $N$ be the midpoints of $AP$ and $BP$ respectively. Lines $MC$ and $ND$ meet at $Q$. Prove that the point $Q$ lies on the lines $OP$, and show that it is independent of the choice of the parallelogram $ABCD$.

Given triangle $ABC$. $D$ is a point on $BC$. $AC$ meets $(ABD)$ again at $E$,and $AB$ meets $(ACD)$ again at $F$. $M$ is the midpoint of $EF$. $BC$ meets $(DEF)$ again at $P$. Prove that $\angle BAP = \angle MAC$.

Let $ABCD$ be a tetrahedron inscribed in a sphere with center $O$, and $G$ and $I$ be its barycenter and incenter respectively. Prove that the following are equivalent: (i) Points $O$ and $G$ coincide. (ii) The four faces of the tetrahedron are congruent. (iii) Points $O$ and $I$ coincide.

Refer to the diagram below. Let $ABCD$ be a cyclic quadrilateral with center $O$. Let the internal angle bisectors of $\angle A$ and $\angle C$ intersect at $I$ and let those of $\angle B$ and $\angle D$ intersect at $J$. Now extend $AB$ and $CD$ to intersect $IJ$ and $P$ and $R$ respectively and let $IJ$ intersect $BC$ and $DA$ at $Q$ and $S$ respectively. Let the midpoints of $PR$ and $QS$ be $M$ and $N$ respectively. Given that $O$ does not lie on the line $IJ$, show that $OM$ and $ON$ are perpendicular.

Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?

Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.

Let $A$ and $B$ different points of a circle $k$ centered at $O$ in such a way such that $AB$ is not a diagonal of $k$. Furthermore, let $X$ be an arbitrary inner point of the segment $AB$. Let $k_1$ be the circle that passes through the points $A$ and $X$, and $A$ is the only common point of $k$ and $k_1$. Similarly, let $k_2$ be the circle that passes through the points $B$ and $X$, and $B$ is the only common point of $k$ and $k_2$. Let $M$ be the second intersection point of $k_1$ and $k_2$. Let $Q$ denote the center of circumscribed circle of the triangle $AOB$. Let $O_1$ and $O_2$ be the centers of $k_1$ and $k_2$. Show that the points $M,O,O_1,O_2,Q$ are on a circle.

Let $A_1A_2A_3A_4A_5A_6$ be a convex hexagon such that $A_iA_{i+2} \parallel A_{i+3}A_{i+5}$ for $i = 1, 2, 3$ (we take $A_{i+6} = A_i$ for each $i$). Segment $A_iA_{i+2}$ intersects segment $A_{i+1}A_{i+3}$ at $B_i$, for $1 \le i \le 6$, as shown. Furthermore, suppose that $\vartriangle A_1A_3A_5 \cong \vartriangle A_4A_6A_2$. Given that $[A_1B_5B_6] = 1$, $[A_2B_6B_1] = 4$, and $[A_3B_1B_2] = 9$ (by $[XY Z]$ we mean the area of $ \vartriangle XY Z$), determine the area of hexagon $B_1B_2B_3B_4B_5B_6$. [img]https://cdn.artofproblemsolving.com/attachments/d/0/1a8997c9eb7dea5223b6805dacd79c10a2cd33.png[/img]

Let $ABCD$ be a rectangle with sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $ A$ intersects $BD$ at the point $H$. We denote by $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the length of segment $MN$.

Let $ABC$ be a triangle with centroid $T$. Denote by $M$ the midpoint of $BC$. Let $D$ be a point on the ray opposite to the ray $BA$ such that $AB = BD$. Similarly, let $E$ be a point on the ray opposite to the ray $CA$ such that $AC = CE$. The segments $T D$ and $T E$ intersect the side $BC$ in $P$ and $Q$, respectively. Show that the points $P, Q$ and $M$ split the segment $BC$ into four parts of equal length.

Let $ABC$ be a triangle, let $B_{1}$ be the midpoint of the side $AB$ and $C_{1}$ the midpoint of the side $AC$. Let $P$ be the point of intersection, other than $A$, of the circumscribed circles around the triangles $ABC_{1}$ and $AB_{1}C$. Let $P_{1}$ be the point of intersection, other than $A$, of the line $AP$ with the circumscribed circle around the triangle $AB_{1}C_{1}$. Prove that $2AP=3AP_{1}$.

The tangents of the circumcircle $\Omega$ of the triangle $ABC$ at points $B$ and $C$ intersect at point $P$. The perpendiculars drawn from point $P$ to lines $AB$ and $AC$ intersect at points$ D$ and $E$ respectively. Prove that the altitudes of the triangle $ADE$ intersect at the midpoint of the segment $BC$.

p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$. Find the number of subsets of $H$. p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result: $0, 1, 2, 3$, then the magician will know that the hidden number is $3$. a. Give an example of your own from the above process. b. Explain mathematically the general form of the process. p3. In a fruit basket there are $20$ apples, $18$ oranges, $16$ mangoes, $10$ pineapples and $6$ papayas. If someone wants to take $10$ pieces from the basket. After that, how many possible compositions of fruit are drawn? p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base $8\sqrt3$ m long and $8$ m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made. p5. What is the remainder of $2012^{2012} + 2014^{2012}$ divided by $2013^2$?

Given a circle with center $O$, such that $A$ is not on the circumcircle. Let $B$ be the reflection of $A$ with respect to $O$. Now let $P$ be a point on the circumcircle. The line perpendicular to $AP$ through $P$ intersects the circle at $Q$. Prove that $AP \times BQ$ remains constant as $P$ varies.