The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$. [b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$. [b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.

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]

The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle{ACB}$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle{CDE}$? $\textbf{(A) }\frac{5\sqrt{2}}{3}\qquad\textbf{(B) }\frac{50\sqrt{3}-75}{4}\qquad\textbf{(C) }\frac{15\sqrt{3}}{8}\qquad\textbf{(D) }\frac{50-25\sqrt{3}}{2}\qquad\textbf{(E) }\frac{25}{6}$

In a triangle $\triangle ABC$ with orthocenter $H$, let $BH$ and $CH$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. If the tangent line to the circumcircle of $\triangle ABC$ passing through $A$ intersects $BC$ at $P$, $M$ is the midpoint of $AH$, and $EF$ intersects $BC$ at $G$, then prove that $PM$ is parallel to $GH$. [i]Proposed by Sreejato Bhattacharya[/i]

In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF \equal{} FG \equal{} GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$. [asy] size(180); pair A, B, C, D, E, F, G, H, J; A = origin; real length = 6; real width = 3.5; B = length*dir(0); C = (length, width); D = width*dir(90); F = length/3*dir(0); G = 2*length/3*dir(0); E = (length/2, width); H = extension(A, C, E, F); J = extension(A, C, E, G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$A$", A, dir(180)); label("$D$", D, dir(180)); label("$B$", B, dir(0)); label("$C$", C, dir(0)); label("$F$", F, dir(270)); label("$E$", E, dir(90)); label("$G$", G, dir(270)); label("$H$", H, dir(140)); label("$J$", J, dir(340)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {5}{2} \qquad \textbf{(B)} \ \frac {35}{12} \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ \frac {7}{2} \qquad \textbf{(E)} \ \frac {35}{8}$

$w_B$ and $w_C$ are excircles of a triangle $ABC$. The circle $w_B'$ is symmetric to $w_B$ with respect to the midpoint of $AC$, the circle $w_C'$ is symmetric to $w_C$ with respect to the midpoint of $AB$. Prove that the radical axis of $w_B'$ and $w_C'$ halves the perimeter of $ABC$.

An equilateral triangle $ABC$ is given, together with a point $P$ inside it. [asy] draw((0,0)--(4,0)--(2,3.464)--(0,0)); draw((1.3, 1.2)--(0,0)); draw((1.3, 1.2)--(2,3.464)); draw((1.3, 1.2)--(4,0)); label("$A$",(0,0),SW); label("$B$",(4,0),SE); label("$C$",(2,3.464),N); label("$P$",(1.3,1.2),S); [/asy] Given that $PA = 3$ cm, $PB = 5$ cm, and $PC = 4$ cm, find the side of the equilateral triangle.

Let $a$ and $b$ be real numbers such that $\sin a+\sin b=\dfrac{\sqrt2}2$ and $\cos a+\cos b=\dfrac{\sqrt6}2$. Find $\sin(a+b)$. $\textbf{(A) }\dfrac12\qquad\textbf{(B) }\dfrac{\sqrt2}2\qquad\textbf{(C) }\dfrac{\sqrt3}2\qquad\textbf{(D) }\dfrac{\sqrt6}2\qquad\textbf{(E) }1$

Quadrilateral $ABCD$ has $AB=BC=CD$, $\angle ABC=70^\circ$, and $\angle BCD=170^\circ$. What is the degree measure of $\angle BAD$? $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 80\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ 95$

A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $\text{(A)}\ 309 \qquad \text{(B)}\ 349 \qquad \text{(C)}\ 369 \qquad \text{(D)}\ 389\qquad \text{(E)}\ 409$

Sketch the curve $$y= \frac{x}{1+x^6 (\sin x)^{2}},$$ and show that $$ \int_{0}^{\infty} \frac{x}{1+x^6 (\sin x)^{2}}\; dx$$ exists.

Calculate the following indefinite integrals. [1] $\int \frac{x}{\sqrt{5-x}}dx$ [2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$ [3] $\int (\sin x+\cos x)^2dx$ [4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$ [5]$\int (\sin x+\sin 2x)^2 dx$

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

Two circles $K_{1}$ and $K_{2}$, centered at $O_{1}$ and $O_{2}$ with radii $1$ and $\sqrt{2}$ respectively, intersect at $A$ and $B$. Let $C$ be a point on $K_{2}$ such that the midpoint of $AC$ lies on $K_{1}$. Find the length of the segment $AC$ if $O_{1}O_{2}=2$

Function $f$ satisfies the equation $f(\cos x) = \cos (17x)$. Prove that it also satisfies the equation $f(\sin x) = \sin (17x)$.

Suppose $a$ and $b$ are real numbers such that \[\lim_{x\to 0}\frac{\sin^2 x}{e^{ax}-bx-1}=\frac{1}{2}.\] Determine all possible ordered pairs $(a, b)$.

Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.

Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

For integer $ n,\ a_n$ is difined by $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\cos x)^ndx$. (1) Find $ a_{\minus{}2},\ a_{\minus{}1}$. (2) Find the relation of $ a_n$ and $ a_{n\minus{}2}$. (3) Prove that $ a_{2n}\equal{}b_n\plus{}\pi c_n$ for some rational number $ b_n,\ c_n$, then find $ c_n$ for $ n<0$.

(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that: \[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \] (2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?

Let $a, b, c, d$ be real numbers satisfying inequality $a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$ holds for arbitrary real number $x$. Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,d$ when that maximum is attained.

Show that:\[\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2 (k\in \mathbb{Z}) \Longleftrightarrow\ |{\tan \alpha}| + |{\cot \alpha}| = 4\]

In $ABC$ with $\angle C = 60^{\circ}$, prove that \[\frac{c}{a} + \frac{c}{b} \ge2.\]

Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.

The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?