An equilateral triangle with integer side length $n$ is subdivided into smaller equilateral triangles of side length $1$ by drawing lines parallel to its sides, as shown in the figure for $n = 4$. [asy] size(5cm); // Function to draw an equilateral triangle with subdivisions and mark vertices void drawTriangleWithDots(pair A, pair B, pair C, int n) { real step = 1.0 / n; // Draw horizontal lines for (int i = 0; i <= n; ++i) { pair start = A + i * step * (C - A); pair end = start + i * step * (B - C); draw(start -- end, gray(0.5)); } // Draw left-leaning diagonal lines for (int i = 0; i <= n; ++i) { pair start = A + i * step * (B - A); pair end = start + (n - i) * step * (C - A); draw(start -- end, gray(0.5)); } // Draw right-leaning diagonal lines for (int i = 0; i <= n; ++i) { pair start = B + i * step * (C - B); pair end = start + (n - i) * step * (A - B); draw(start -- end, gray(0.5)); } // Mark dots at all vertices for (int i = 0; i <= n; ++i) { for (int j = 0; j <= i; ++j) { pair vertex = A + i * step * (C - A) + j * step * (B - C); dot(vertex, black); } } // Draw the outer triangle draw(A -- B -- C -- cycle, black+linewidth(1)); } // Main triangle vertices pair A = (0, 0); pair B = (4, 0); pair C = (2, 3.464); // Height = sqrt(3)/2 * side length // Subdivisions int n = 4; // Draw the subdivided equilateral triangle with dots drawTriangleWithDots(A, B, C, n); [/asy] Consider the set $A$ consisting of all points that are vertices of any of these smaller triangles. A [i]subtriangle[/i] is defined as any equilateral triangle whose three vertices belong to the set $A$ and whose three sides lie along the lines of the initial subdivision. We wish to color all points in $A$ either red or blue such that no subtriangle has all three vertices of the same color. Let $C(n)$ denote the number of such valid colorings for each positive integer $n$. Calculate, in terms of $n$, the value of $C(n)$.

Set $p\ge 5$ be a prime number and $n$ be a natural number. Let $f$ be a function $ f: \mathbb{Z_{ \neq }}_0 \rightarrow \mathbb{ N }_0 $ satisfy the following conditions: i) For all sequences of integers satisfy $ a_i \not\in \{0, 1\} $, and $ p $ $\not |$ $ a_i-1 $, $ \forall $ $ 1 \le i \le p-2 $,\\ $$ \displaystyle \sum^{p-2}_{i=1}f(a_i)=f(a_1a_2 \cdots a_{p-2}) $$ ii) For all coprime integers $ a $ and $ b $, $ a \equiv b \pmod p \Rightarrow f(a)=f(b) $ iii) There exist $k \in \mathbb{Z}_{\neq 0} $ that satisfy $ f(k)=n $ Prove that the number of such functions is $ d(n) $, where $ d(n) $ denotes the number of divisors of $ n $.

Inside parallelogram $ABCD$ is point $P$, such that $PC = BC$. Show that line $BP$ is perpendicular to line which connects middles of sides of line segments $AP$ and $CD$.

Let $ABC$ be a triangle with incircle touching $BC, CA, AB$ at $D, E, F,$ respectively. Let $O$ and $M$ be its circumcenter and midpoint of $BC.$ Suppose that circumcircles of $AEF$ and $ABC$ intersect at $X$ for the second time. Assume $Y \neq X$ is on the circumcircle of $ABC$ such that $OMXY$ is cyclic. Prove that circumcenter of $DXY$ lies on $BC.$ [i]Proposed by tenplusten.[/i]

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

Compute the number of ordered pairs $(m,n)$ of positive integers that satisfy the equation $\text{lcm}(m,n)+\gcd(m,n)=m+n+30$. [i]Proposed by Ankit Bisain[/i]

Given 2015 balls. Astri and Budi will play a game. At first, Astri will choose two different numbers $a$ and $b$ from the set $S = \{ 1, 2, 3, \dots, 30 \}$. Budi will then choose another 2 different numbers $c$ and $d$ from the remaining 28 numbers in set $S$. By taking turns, starting from Astri, they take balls with the following rules: (1) Astri could only take $a$ or $b$ balls. (2) Budi could only take $c$ or $d$ balls. until someone couldn't take any balls satisfying the condition given (and that person will lose). Prove that Budi could choose $c,d$ such that he has a strategy to ensure his victory on this game.

Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that \[ f(x)+f(y)+f(z)=1 \] holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$.

A quadratic polynomial $f(x)$ is called sparse if its degree is exactly 2 , if it has integer coefficients, and if there exists a nonzero polynomial $g(x)$ with integer coefficients such that $f(x) g(x)$ has degree at most 3 and $f(x) g(x)$ has at most two nonzero coefficients. Find the number of sparse quadratics whose coefficients lie between 0 and 10, inclusive.

Prove that for any real numbers $ x_1, x_2, x_3, \ldots, x_n $ the inequality is true $$ x_1x_2x_3\ldots x_n \leq \frac{x_1^2}{2} + \frac{x_2^4}{4} + \frac{x_3^8}{8} + \ldots + \frac{x_n^{2^ n}}{2^n} + \frac{1}{2^n}$$

A [i]coloring[/i] of the set of integers greater than or equal to $1$, must be done according to the following rule: Each number is colored blue or red, so that the sum of any two numbers (not necessarily different) of the same color is blue. Determine all the possible [i]colorings[/i] of the set of integers greater than or equal to $1$ that follow this rule.

A fair coin is flipped every second and the results are recorded with $1$ meaning heads and $0$ meaning tails. What is the probability that the sequence $10101$ occurs before the first occurance of the sequence $010101$?

Let $ABC$ be a triangle with $AB < AC$. As shown below, $T$ is the point on $\overline{BC}$ such that $\overline{AT}$ is tangent to the circumcircle of $\triangle{}ABC$. Additionally, $H$ and $O$ are the orthocenter and circumcenter of $\triangle{}ABC$, respectively. Suppose that $\overline{CH}$ passes through the midpoint of $\overline{AT}$. Prove that $\overline{AO}$ bisects $\overline{CH}$. [asy] size(8cm); pair A = dir(132.5); pair B = dir(200); pair C = dir(340); draw(A--B--C--cycle, black); draw(circumcircle(A, B, C), black); pair O = circumcenter(A, B, C); pair U = 2*C*A/(C+A); pair V = 2*A*B/(A+B);pair T = extension(U, V, B, C); draw(A--T); draw(T--B);pair X = (T+A)/2; pair H = (A+B+C); draw(A--H); pair Y = (H+C)/2; draw(H--X, dashed); draw(C--H); draw(O--Y, dashed); draw(A--O); dot("$A$", A, dir(A)); dot("$B$", B, SW); dot("$C$", C, dir(C)); dot("$O$", O, NE); dot("$T$", T, dir(T)); dot("$H$", H, SW); dot("$X$", X, NW); dot("$Y$", Y, SW); [/asy]

A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.

Let $n$ be a positive integer. Consider the sum $x_1y_1 + x_2y_2 +\cdots + x_ny_n$, where that values of the variables $x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n$ are either 0 or 1. Let $I(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum of the number is odd, and let $P(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum is an even number. Show that: \[ \frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1} \]

Let $ABC$ be an equilateral triangle and let $P_0$ be a point outside this triangle, such that $\triangle{AP_0C}$ is an isoscele triangle with a right angle at $P_0$. A grasshopper starts from $P_0$ and turns around the triangle as follows. From $P_0$ the grasshopper jumps to $P_1$, which is the symmetric point of $P_0$ with respect to $A$. From $P_1$, the grasshopper jumps to $P_2$, which is the symmetric point of $P_1$ with respect to $B$. Then the grasshopper jumps to $P_3$ which is the symmetric point of $P_2$ with respect to $C$, and so on. Compare the distance $P_0P_1$ and $P_0P_n$. $n \in N$.