Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

For each natural number $n\ge3$, let $m(n)$ be the maximum number of points inside or on the sides of a regular $n$-agon of side $1$ such that the distance between any two points is greater than $1$. Prove that $m(n)\ge n$ for $n>6$.

Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively. 30 points

[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

Investigate the asymptotic behaviour of the solutions $y$ of the equation $x^5 + x^2y^2 = y^6$ that tend to zero as $x\to0$.

A region $ S$ in the complex plane is defined by \[ S \equal{} \{x \plus{} iy: \minus{} 1\le x\le1, \minus{} 1\le y\le1\}.\] A complex number $ z \equal{} x \plus{} iy$ is chosen uniformly at random from $ S$. What is the probability that $ \left(\frac34 \plus{} \frac34i\right)z$ is also in $ S$? $ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac23\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac79\qquad \textbf{(E)}\ \frac78$

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

The number of $ x$-intercepts on the graph of $ y \equal{} \sin(1/x)$ in the interval $ (0.0001,0.001)$ is closest to $ \textbf{(A)}\ 2900 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3100 \qquad \textbf{(D)}\ 3200 \qquad \textbf{(E)}\ 3300$

In a non-isosceles triangle $ABC$ the bisectors of angles $A$ and $B$ are inversely proportional to the respective sidelengths. Find angle $C$.

Let $\alpha$ and $\beta$ be reals. Find the least possible value of $$(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.$$

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is [asy] size(170); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=(1.5,3*sqrt(3)/2), C=(3,0), D=(1,0), P=B+1.6*dir(B--A), Q=C+1.2*dir(C--A); draw(B--P--D--B^^P--Q--D--C--Q); draw(Q--A--P, linetype("4 4")); label("$A$", A, N); label("$B$", B, W); label("$C$", C, E); label("$A'$", D, S); label("$P$", P, W); label("$Q$", Q, E); [/asy] $ \textbf{(A)}\ \frac{8}{5}\qquad\textbf{(B)}\ \frac{7}{20}\sqrt{21}\qquad\textbf{(C)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{13}{8}\qquad\textbf{(E)}\ \sqrt{3} $

Given a triangle $ABC$ with $BC=a$, $CA=b$, $AB=c$, $\angle BAC = \alpha$, $\angle CBA = \beta$, $\angle ACB = \gamma$. Prove that $$ a \sin(\beta-\gamma) + b \sin(\gamma-\alpha) +c\sin(\alpha-\beta) = 0.$$

$ ABC$ is an acute-angled triangle. $ M$ is the midpoint of $ BC$ and $ P$ is the point on $ AM$ such that $ MB \equal{} MP$. $ H$ is the foot of the perpendicular from $ P$ to $ BC$. The lines through $ H$ perpendicular to $ PB$, $ PC$ meet $ AB, AC$ respectively at $ Q, R$. Show that $ BC$ is tangent to the circle through $ Q, H, R$ at $ H$. [i]Original Formulation: [/i] For an acute triangle $ ABC, M$ is the midpoint of the segment $ BC, P$ is a point on the segment $ AM$ such that $ PM \equal{} BM, H$ is the foot of the perpendicular line from $ P$ to $ BC, Q$ is the point of intersection of segment $ AB$ and the line passing through $ H$ that is perpendicular to $ PB,$ and finally, $ R$ is the point of intersection of the segment $ AC$ and the line passing through $ H$ that is perpendicular to $ PC.$ Show that the circumcircle of $ QHR$ is tangent to the side $ BC$ at point $ H.$

Let $a_1,a_2,\ldots ,a_9$ be any non-negative numbers such that $a_1=a_9=0$ and at least one of the numbers is non-zero. Prove that for some $i$, $2\le i\le 8$, the inequality $a_{i-1}+a_{i+1}<2a_i$ holds. Will the statement remain true if we change the number $2$ in the last inequality to $1.9$?

In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$. Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.

Prove that for an arbitrary triangle $ABC$ : $sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} < \frac{1}{4}$.

Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true: If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]

Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with \[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\] Find $ \det A$.

Triangle $ABC$ is right-angled at $C$ with $AC = b$ and $BC = a$. If $d$ is the length of the altitude from $C$ to $AB$, prove that $\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{d^2}$

Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$ Answer the questions below. (1) Find $I_n.$ (2) Find $\sum_{n=1}^{\infty} I_n.$

Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that \[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]