The line $T$ is tangent to the circumcircle of acute triangle $ABC$ at $B$. Let $K$ be the projection of the orthocenter of triangle $ABC$ onto line $T$ ($K$ is the root of the perpendicular from the orthocenter to $S$). Let $L$ be the midpoint of side $AC$. Show that the triangle $BKL$ is isosceles.
The squares $BCDE$, $CAFG$, and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is isosceles and $\angle PAQ =\frac{\pi}{2} $.
Given triangle $ABC$, construct equilateral triangle $ABC_1$, $BCA_1$, $CAB_1$ on the outside of $ABC$. Let $P, Q$ denote the midpoints of $C_1A_1$ and $C_1B_1$ respectively. Let $R$ be the midpoint of $AB$.
Prove that triangle $PQR$ is isosceles.
(Napolean's Triangle) Given triangle $ABC$, construct an equilateral triangle on the outside of each of the sides. Let $P, Q, R$ be the centroids of these equilateral triangles, prove that triangle $PQR$ is equilateral.
(Simson Line) Let $Z$ be a point on the circumcircle of triangle $ABC$ and $P, Q, R$ be the feet of perpendiclars from $Z$ to $BC, AC, AB$ respectively. Prove that $P, Q, R$ are collinear. (This line is called the Simson line of triangle $ABC$ from $Z$.)
Given cyclic quadrilateral $ABCD$, let $P$ and $Q$ be the reflection of $C$ across lies $AB$ and $AD$ respectively.
Prove that PQ passes through the orthocentre of triangle $ABD$.
Let $W_1W_2W_3$ be a triangle with circumcircle $S$, and let $A_1, A_2, A_3$ be the midpoints of $W_2W_3, W_1W_3,
W_1W_2$ respectively. From Ai drop a perpendicular to the line tangent to $S$ at $W_i$. Prove that these perpendicular lines are concurrent and identify this point of concurrency.
Let $A_0A_1A_2A_3A_4A_5A_6$ be a regular 7-gon. Prove that $$\frac{1}{A_0A_1} = \frac{1}{A_0A_2}+\frac{1}{A_0A_3}$$
Given point $P_0$ in the plane of triangle $A_1A_2A_3$. Denote $A_s = A_{s-3}$, for $s > 3$. Construct points $P_1; P_2; \cdots$ sequentially such that point $P_{k+1}$ is $P_k$ rotated $120^\circ$ counter-clockwise around $A_{k+1}$. Prove that if $P_{1986} = P_0$ then triangle $A_1A_2A_3$ is isosceles.
Point $H$ is the orthocenter of triangle $ABC$. Points $D, E$ and $F$ lie on the circumcircle of triangle $ABC$ such that $AD\parallel BE\parallel CF$. Points $S, T,$ and $U$ are the respective reflections of $D, E, F$ across the
lines $BC, CA$ and $AB$. Prove that $S, T, U, H$ are cyclic.
Let $ABCD$ be a cyclic quadrilateral. Let $P, Q, R$ be the feet of the perpendiculars from $D$ to the lines $BC, CA$ and $AB$ respectively. Show that $PQ = QR$ iff the bisectors of $\angle ABC$ and $\angle ADC$ meet on $AC$.
Let $O$ be the circumcentre of triangle $ABC$. A line through $O$ intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Let $S$ and $R$ be the midpoints of $BN$ and $CM$, respectively. Prove that $\angle ROS = \angle BAC$.
Let $ABCD$ be a convex quadrilateral for which $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral and pointing outward. Let $O_1, O_2, O_3, O_4$ be the centres of the triangles constructed on $AB, BC, CD,$ and $DA$ respectively. Prove that lines $O_1O_3$ and $O_2O_4$ are perpendicular.
Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle BAP = \angle CAQ$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcentre of triangle $BCR$. Prove that $AO \perp PQ$.
Express the golden ratio using a continued fraction.
Find the rational number $p/q$ closest to $\sqrt{\pi}$ wich $q \le 25$.
Find the number $x = [1, 2, 3, 1, 2, 3, \cdots]$. (continued fraction)
Simplify $\displaystyle\frac{2^2-2}{2^2+2}\cdot\displaystyle\frac{3^2-3}{3^2+3}\cdots\displaystyle\frac{10^2-2}{10^2+10}$
Let $a$, $b$ be real numbers such that $ab=-1$, $a+b=3$, compute $a^3+b^3$.
Compute $\Large(\sqrt{6+4\sqrt{2}} + \sqrt{6-4\sqrt{2}}\Large)^2$
Find the roots of $27x^3 + 9x^2 -30+8$
Denote $a$ and $b$ the roots of $(x-2)(x+4)+(x-3)(x+5)-(x-2)(x+5)=0$. Computer $a^3 + b^3 + \frac{11}{(a-1)(b-1)}$
Find all nonnegative integers $x$ and $y$ such that $x^3+y^3 = (x+y)^2$.
Let $a$, $b$, $c$ be distinct nonzero real numbers, such that $$a+\frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$$
Prove that $|abc|=1$.