Let $ Q$ be the centre of the inscribed circle of a triangle $ ABC.$ Prove that for any point $ P,$ \[ a(PA)^2 \plus{} b(PB)^2 \plus{} c(PC)^2 \equal{} a(QA)^2 \plus{} b(QB)^2 \plus{} c(QC)^2 \plus{} (a \plus{} b \plus{} c)(QP)^2, \] where $ a \equal{} BC, b \equal{} CA$ and $ c \equal{} AB.$

Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\hat{A}$.

Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to $ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

Let $ABC$ be an acute triangle with $\angle B > \angle C$. On the circle $\mathcal{C}(O, R)$ circumscribed to this triangle points $D, E, J, K, S$ are chosen such that $A, E, J$ and $K$ are on the same side of the line $BC$, the diameter $DE$ is perpendicular on the chord $BC$, $S\in \overarc{EK},\overarc{AE}=\overarc{BJ}=\overarc{CK}=\dfrac{1}{4}\overarc{CE}$ . Let $\{F\}=AC\cap DE, \{M\}=BK\cap AD, \{P\}=BK\cap AC$ and $\{Q\}=CJ\cap BF$. If $\angle SMK =30^{\circ}$ and $\angle AQP = 90^{\circ}$, show that the line $MS$ is tangent to the circumscribed circle of triangle $AOF$.

Determine the number of positive odd and even factor of $ 5^6\minus{}1$.

Real numbers $x,y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x,y$, and $z$ are within $1$ unit of each other is greater than $\tfrac{1}{2}$. What is the smallest possible value of $n$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

[b]2A. [/b] Two triangles $ABC$ and $ABD$ share a common side. $ABC$ is drawn such that its entire area lies inside the larger triangle $ABD$. If $AB = 20$, side $AD$ meets side $AB$ at a right angle, and point $C$ is between points $A$ and $D$, then find the area outside of triangle $ABC$ but within $ABD$, given that both triangles have integral side lengths and $AB$ is the smallest side of either triangle. $ABC$ and $ABD$ are both primitive right triangles. [i] (1 point)[/i] [b]2B.[/b] Find the sum of all positive integral factors of the correct answer to [b]2A[/b]. [i](2 points)[/i] [b]2C.[/b] Let $B$ be the sum of the digits of the correct answer to [b]2B[/b] above. If the solution to the functional equation $21*f(x) - 7*f(1/x) = Bx$ is of the form $(Ax^2 + C) / Dx$, find $C$, given that $A$, $C$, and $D$ are relatively prime (they don’t share a common prime factor). [i](3 points)[/i] [hide=ANSWER KEY]2A.780 2B. 2352 2C. 3[/hide]

Let $K$ be a point inside an acute triangle $ ABC $, such that $ BC $ is a common tangent of the circumcircles of $ AKB $ and $ AKC$. Let $ D $ be the intersection of the lines $ CK $ and $ AB $, and let $ E $ be the intersection of the lines $ BK $ and $ AC $ . Let $ F $ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$. The circumcircle of $ABC$ and the circle $k$ with centre $ F$ and radius $FD$ intersect at points $P$ and $Q$. Prove that the segment $PQ$ is a diameter of $k$.

The points $ P, Q$ are given on the plane, which are the points of intersection of the angle bisector $AL$ of some triangle $ABC$ with an inscribed circle, and the point $W$ is the intersection of the angle bisector $AL$ with a circumscribed circle other than the vertex $A$. a) Find the geometric locus of the possible location of the vertex $A$ of the triangle $ABC$. b) Find the geometric locus of the possible location of the vertex $B$ of the triangle $ABC$.

Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.

Let $ABC$ be a triangle and $AB<AC<BC$. Let $D,E$ be points on the side $BC$ and the line $AB$, respectively ($A$ is between $B,E$) such that $BD=BE=AC$. The circumcircle of $\Delta BED$ meets the side $AC$ at $P$ and $BP$ meets the circumcircle of $\Delta ABC$ at $Q$. Prove that: \[ AQ+CQ=BP. \]

Let $ ABCD$ be a rhombus with $ \angle BAD \equal{} 60^{\circ}$. Points $ S$ and $ R$ are chosen inside the triangles $ ABD$ and $ DBC$, respectively, such that $ \angle SBR \equal{} \angle RDS \equal{} 60^{\circ}$. Prove that $ SR^2\geq AS\cdot CR$.

A pentagon has vertices labelled $A, B, C, D, E$ in that order counterclockwise, such that $AB$, $ED$ are parallel and $\angle EAB = \angle ABD = \angle ACD = \angle CDA$. Furthermore, suppose that$ AB = 8$, $AC = 12$, $AE = 10$. If the area of triangle $CDE$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are integers so that $b$ is square free, and $a, c$ are relatively prime, find $a + b + c$.

Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane. Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.

Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC}$, respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two non square rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD$. Find $\frac{AE}{EB} + \frac{EB}{AE}$.

A $2 \times 4 \times 8$ rectangular prism and a cube have the same volume. What is the difference between their surface areas?

Let $ABC$ be a triangle and $AD,BE,CF$ be cevians concurrent at a point $P$. Suppose each of the quadrilaterals $PDCE,PEAF$ and $PFBD$ has both circumcircle and incircle. Prove that $ABC$ is equilateral and $P$ coincides with the center of the triangle.

Show that any positive rational number can be represented as the sum of three positive rational cubes.

In an acute triangle $ABC$, an arbitrary point $P$ is chosen on the altitude $AH$. The points $E$ and $F$ are the midpoints of $AC$ and $AB$, respectively. The perpendiculars from $E$ on $CP$ and from $F$ on $BP$ intersect at the point $K$. Show that $KB = KC$.

For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$

Source: 1976 Euclid Part B Problem 1 ----- Triangle $ABC$ has $\angle{B}=30^{\circ}$, $AB=150$, and $AC=50\sqrt{3}$. Determine the length of $BC$.

Let $AD,BF,CE$ be altitudes of triangle $ABC$.$Q$ is a point on $EF$ such that $QF=DE$ and $F$ is between $E,Q$.$P$ is a point on $EF$ such that $EP=DF$ and $E$ is between $P,F$.Perpendicular bisector of $DQ$ intersect with $AB$ at $X$ and perpendicular bisector of $DP$ intersect with $AC$ at $Y$.Prove that midpoint of $BC$ lies on $XY$.

Let $ABCD$ be a rectangle with $BC=24.$ Point $X$ lies inside the rectangle such that $\angle{AXB}=90^\circ.$ Given that triangles $\triangle{AXD}$ and $\triangle{BXC}$ are both acute and have circumradii $13$ and $15,$ respectively, compute $AB.$

Let $ABCD$ be a parallelogram. Points $E, F$ lie on the sides $AB, CD$ respectively, such that $\angle EDC = \angle FBC$ and $\angle ECD = \angle FAD$. Prove that $AB \geq 2BC$. [i]Proposed by Pouria Mahmoudkhan Shirazi - Iran[/i]