Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged. Find the maximum possible contingents such that the above condition still holds? [i]Raymond Christopher Sitorus, Singapore[/i]

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

Let $ABC$ be a triangle in which $\angle ABC = 45^o$ and $\angle BAC > 90^o$. Let $O$ be the midpoint of the side $[BC]$. Consider the point $M \in (AC)$ such that $\angle COM =\angle CAB$. Perpendicular from $M$ on $AC$ intersects line $AB$ at point $P$. a) Find the measure of the angle $\angle BCP$. b) Show that if $\angle BAC = 105^o$, then $PB = 2MO$.

Given distinct straight lines $ OA$ and $ OB$. From a point in $ OA$ a perpendicular is drawn to $ OB$; from the foot of this perpendicular a line is drawn perpendicular to $ OA$. From the foot of this second perpendicular a line is drawn perpendicular to $ OB$; and so on indefinitely. The lengths of the first and second perpendiculars are $ a$ and $ b$, respectively. Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is: $ \textbf{(A)}\ \frac {b}{a \minus{} b} \qquad \textbf{(B)}\ \frac {a}{a \minus{} b} \qquad \textbf{(C)}\ \frac {ab}{a \minus{} b} \qquad \textbf{(D)}\ \frac {b^2}{a \minus{} b} \qquad \textbf{(E)}\ \frac {a^2}{a \minus{} b}$

Determine all integers $n\ge 2$ such that for any two infinite sequences of positive integers $a_1<a_2< \cdots $ and $b_1, b_2, \cdots$, such that $a_i\mid a_j$ for all $i<j$, there always exists a real number $c$ such that $$\lfloor{ca_i}\rfloor \equiv b_i \pmod {n}$$ for all $i\ge 1$. [i]Proposed by Wong Jer Ren & Ivan Chan Kai Chin[/i]

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.

\[ a\plus{}4b\plus{}9c\plus{}16d\plus{}25e\plus{}36f\plus{}49g\equal{}1\] \[ 4a\plus{}9b\plus{}16c\plus{}25d\plus{}36e\plus{}49f\plus{}64g\equal{}12\] \[ 9a\plus{}16b\plus{}25c\plus{}36d\plus{}49e\plus{}64f\plus{}81g\equal{}123\] Determine the value of $ 16a\plus{}25b\plus{}36c\plus{}49d\plus{}64e\plus{}81f\plus{}100g$.

Given a positive integer $n$. One of the roots of a quadratic equation $x^{2}-ax +2 n = 0$ is equal to $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}$. Prove that $2\sqrt{2n}\le a\le 3\sqrt{n}$

Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$

Two sums, each consisting of $n$ addends , are shown below: $S = 1 + 2 + 3 + 4 + ...$ $T = 100 + 98 + 96 + 94 +...$ . For what value of $n$ is it true that $S = T$ ?

Let $ a_{1},a_{2},...,a_{n} $ be real positive numbers and $ k>m, k,m $ natural numbers. Prove that $(n-1)(a_{1}^m +a_{2}^m+...+a_{n}^m)\leq\frac{a_{2}^k+a_{3}^k+...+a_{n}^k}{a_{1}^{k-m}}+\frac{a_{1}^k+a_{3}^k+...+a_{n}^k}{a_2^{k-m}}+...+\frac{a_{1}^k+a_{2}^k+...+a_{n-1}^k}{a_{n}^{k-m}} $

Evaluate $ \int_0^1 \frac{2e^{2x}\plus{}xe^x\plus{}3e^x\plus{}1}{(e^x\plus{}1)^2(e^x\plus{}x\plus{}1)^2}\ dx$.

Find all positive integers $ a, b, c, d $ so that $ a^2+b^2+c^2+d^2=13 \cdot 4^n $