The function $ F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $ n \geq 0,$ (i) $ F(4n) \equal{} F(2n) \plus{} F(n),$ (ii) $ F(4n \plus{} 2) \equal{} F(4n) \plus{} 1,$ (iii) $ F(2n \plus{} 1) \equal{} F(2n) \plus{} 1.$ Prove that for each positive integer $ m,$ the number of integers $ n$ with $ 0 \leq n < 2^m$ and $ F(4n) \equal{} F(3n)$ is $ F(2^{m \plus{} 1}).$

Given a prime number $p\ge 3,$ and an integer $d \ge 1$. Prove that there exists an integer $n\ge 1,$ such that $\gcd(n,d) = 1,$ and the product \[P=\prod\limits_{1 \le i < j < p} {({i^{n + j}} - {j^{n + i}})} \text{ is not divisible by } p^n.\]

Given that the expected amount of $1$s in a randomly selected $2021$-digit number is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Hannah Shen[/i]

Points $ A,X,D$ lie on a line in this order, point $ B$ is on the plane such that $ \angle ABX>120^{\circ}$, and point $ C$ is on the segment $ BX$. Prove the inequality: $ 2AD \ge \sqrt{3} (AB\plus{}BC\plus{}CD)$.

Determine all infinite sequences of nonnegative integers $a_1,a_2,\ldots$ such that: 1. Every positive integer appears in the sequence at least once, and; 2. $a_i$ is the smallest integer $j$ such that $a_{j+2}=i$, for all $i\ge 1$. [i](Proposed by Ho Janson)[/i]

Let (i) $a_1,a_2,a_3$ is an arithmetic progression and $a_1+a_2+a_3=18$ (ii) $b_1,b_2,b_3$ is a geometric progression and $b_1b_2b_3=64$ If $a_1+b_1,a_2+b_2,a_3+b_3$ are all positive integers and it is a ageometric progression, then find the maximum value of $a_3$.

Find all $x$ such that $ 6^x + 27^{x-1} = 8^x - 1 $.

Let $ABC$ be an isosceles triangle, $AB = AC, \angle A = 20^{\circ}$. Let $D$ be a point on $AB$, and $E$ a point on $AC$ such that $\angle ACD = 20^{\circ}$ and $\angle ABE = 30^{\circ}$. What is the measure of the angle $\angle CDE$?

Prove that if $ x,y$ are nonnegative real numbers with $ x\plus{}y\equal{}2$, then: $ x^2 y^2 (x^2\plus{}y^2) \le 2$.

Tangents $ l_1$ and $ l_2$ common to circles $ c_1$ and $ c_2$ intersect at point $ P$, whereby tangent points remain to different sides from $ P$ on both tangent lines. Through some point $ T$, tangents $ p_1$ and $ p_2$ to circle $ c_1$ and tangents $ p_3$ and $ p_4$ to circle $ c_2$ are drawn. The intersection points of $ l_1$ with lines $ p_1, p_2, p_3, p_4$ are $ A_1, B_1, C_1, D_1$, respectively, whereby the order of points on $ l_1$ is: $ A_1, B_1, P, C_1, D_1$. Analogously, the intersection points of $ l_2$ with lines $ p_1, p_2, p_3, p_4$ are $ A_2, B_2, C_2, D_2$, respectively. Prove that if both quadrangles $ A_1A_2D_1D_2$ and $ B_1B_2C_1C_2$ are cyclic then radii of $ c_1$ and $ c_2$ are equal.

Let $\Gamma_1$ and $\Gamma_2$ be two circles externally tangent to each other at $N$ that are both internally tangent to $\Gamma$ at points $U$ and $V$ , respectively. A common external tangent of $\Gamma_1$ and $\Gamma_2$ is tangent to $\Gamma_1$ and $\Gamma_2$ at $P$ and $Q$, respectively, and intersects $\Gamma$ at points $X$ and $Y$ . Let $M$ be the midpoint of the arc $XY$ that does not contain $U$ and $V$ . Let $Z$ be on $\Gamma$ such $MZ \perp NZ$, and suppose the circumcircles of $QVZ$ and $PUZ$ intersect at $T\ne Z$. Find, with proof, the value of $T U + T V$ , in terms of $R$, $r_1$, and $r_2$, the radii of $\Gamma$, $\Gamma_1$, and $\Gamma_2$, respectively.

Let $ABC$ be a triangle with $AB < AC$ and circumcenter $O$. The angle bisector of $\angle BAC$ meets the side $BC$ at $D$. The line through $D$ perpendicular to $BC$ meets the segment $AO$ at $X$. Furthermore, let $Y$ be the midpoint of segment $AD$. Prove that points $B, C, X, Y$ are concyclic.

The sides of a convex regular polygon of $L + M + N$ sides are to be given draw in three colors: $L$ of them with a red stroke, $M$ with a yellow stroke, and $N$ with a blue. Express, through inequalities, the necessary and sufficient conditions so that there is a solution (several, in general) to the problem of doing it without leaving two adjacent sides drawn with the same color.

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at a point $X$. The circumcircles of triangles $ABX$ and $CDX$ meet at a point $Y$ (apart from $X$). Let $O$ be the center of the circumcircle of the quadrilateral $ABCD$. Assume that the points $O$, $X$, $Y$ are all distinct. Show that $OY$ is perpendicular to $XY$.

Let $p \equiv 3 \,(\textrm{mod}\, 4)$ be a prime and $\theta$ some angle such that $\tan(\theta)$ is rational. Prove that $\tan((p+1)\theta)$ is a rational number with numerator divisible by $p$, that is, $\tan((p+1)\theta) = \frac{u}{v}$ with $u, v \in \mathbb{Z}, v >0, \textrm{mdc}(u, v) = 1$ and $u \equiv 0 \,(\textrm{mod}\,p) $.

A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$, she takes a step based on the following rules: (a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$; (b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$. How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x,y)$ to $(x+1,y))$?

On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.

Triangle $ABC$ is equilateral with side length $12$. Point $D$ is the midpoint of side $\overline{BC}$. Circles $A$ and $D$ intersect at the midpoints of side $AB$ and $AC$. Point $E$ lies on segment $\overline{AD}$ and circle $E$ is tangent to circles $A$ and $D$. Compute the radius of circle $E$. [i]2015 CCA Math Bonanza Individual Round #5[/i]

A rectangular field is 80 m long and 60 m wide. If fence posts are placed at the corners and are 10 m apart along the 4 sides of the field, how many posts are needed to completely fence the field? $\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

Let $AH_a$ and $BH_b$ be altitudes, $AL_a$ and $BL_b$ be angle bisectors of a triangle $ABC$. It is known that $H_aH_b // L_aL_b$. Is it necessarily true that $AC = BC$? (B. Frenkin)

Alice and Bob play the following game: Alice picks a set $A = \{1, 2, ..., n \}$ for some natural number $n \ge 2$. Then, starting from Bob, they alternatively choose one number from the set $A$, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by $1$ from an already chosen number. The game ends when all numbers from the set $A$ are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy. Proposed by [i]Demetres Christofides, Cyprus[/i]

Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.

Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$ [b](a)[/b] Find the general fomular for $a_{n}$ [b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences

A positive integer \(n\) is called perfect if the sum of its positive divisors \(\sigma(n)\) is twice \(n\), that is, \(\sigma(n) = 2n\). For example, \(6\) is a perfect number since the sum of its positive divisors is \(1 + 2 + 3 + 6 = 12\), which is twice \(6\). Prove that if \(n\) is a positive perfect integer, then: \[ \sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1} \] where the sums are taken over all prime divisors \(p\) of \(n\).

In triangle $ABC$, $Q$ and $R$ are points on segments $AC$ and $AB$, respectively, and $P$ is the intersection of $CR$ and $BQ$. If $AR=RB=CP$ and $CP=PQ$, find $ \angle BRC $.