We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. We call a nonnegative integer $r$-Fibonacci number if it is a sum of $r$ (not necessarily distinct) Fibonacci numbers. Show that there infinitely many positive integers that are not $r$-Fibonacci numbers for any $r, 1 \leq r\leq 5.$

Compute the sum of all positive integers less than $100$ that do not have consecutive $1$s in their binary representation.

Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

Prove that for every positive integer $m$, every prime $p$ and every positive integer $j \le p^{m-1}$, $p^m$ divides $${p^m \choose p^j }- {p^{m-1} \choose j}$$

If the real function $f(x)=\cos x+\sum_{i=1}^{n}\cos(a_ix)$ is periodic, prove that $a_i,i\in\{1,2,...,n\}$, are rational numbers.

Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$.

The point $H$ is the orthocentre of a triangle $ABC$, and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point.

A cube with sides of length 3cm is painted red and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm. If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that are not painted at all, b the number painted on one sides, c the number painted on two sides, and d the number painted on three sides, determine the value a-b-c+d.

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

A disk in a record turntable makes $100$ revolutions per minute and it plays during $24$ minutes and $30$ seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from $29$cm to $11.5$cm. Compute the length of the recorded line.

Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).

Among the $100$ constants $a_1,a_2,a_3,...,a_{100}$, there are $39$ equal to $-1$ and $61$ equal to $+1$. Find the sum of all the products $a_ia_j$, where $1\le i < j \le 100$.

Let $\vartriangle ABC$ be an acute triangle with $AB > AC$. Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$. Let $X$ be the intersection of $PQ$ and $BC$. Let the intersection of the circumcircles of triangle $\vartriangle AXC$ and triangle $\vartriangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$.

Consider the set \[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\] [list=a] [*]Prove that every integer $x \geq 2$ can be written as the product of one or more elements of $A$, which are not necessarily different. [*]For every integer $x \geq 2$ let $f(x)$ denote the minimum integer such that $x$ can be written as the product of $f(x)$ elements of $A$, which are not necessarily different. Prove that there exist infinitely many pairs $(x,y)$ of integers with $x\geq 2$, $y \geq 2$, and \[f(xy)<f(x)+f(y).\] (Pairs $(x_1,y_1)$ and $(x_2,y_2)$ are different if $x_1 \neq x_2$ or $y_1 \neq y_2$). [/list]

If $ y \equal{} f(x) \equal{} \frac {x \plus{} 2}{x \minus{} 1}$, then it is incorrect to say: $ \textbf{(A)}\ x \equal{} \frac {y \plus{} 2}{y \minus{} 1} \qquad\textbf{(B)}\ f(0) \equal{} \minus{} 2 \qquad\textbf{(C)}\ f(1) \equal{} 0 \qquad\textbf{(D)}\ f( \minus{} 2) \equal{} 0$ $ \textbf{(E)}\ f(y) \equal{} x$

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

From $1,2,\cdots,14$, take out three numbers $a_1<a_2<a_3$, satisfying that $a_2-a_1\geq3,a_3-a_2\geq3$. Then the number of different ways of taking out numbers is________.

Let $O$ be the circumcenter of acutangle triangle $ABC$ and let $A_1$ be some point in the smallest arc $BC$ of the circumcircle of $ABC$. Let $A_2$ and $A_3$ points on sides $AB$ and $AC$, respectively, such that $\angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$. Prove that the line $A_2A_3$ passes through the orthocenter of $ABC$.

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ a function having the property that $$ \left| f(x+y)+\sin x+\sin y \right|\le 2, $$ for all real numbers $ x,y. $ [b]a)[/b] Prove that $ \left| f(x) \right|\le 1+\cos x, $ for all real numbers $ x. $ [b]b)[/b] Give an example of what $ f $ may be, if the interval $ \left( -\pi ,\pi \right) $ is included in its [url=https://en.wikipedia.org/wiki/Support_(mathematics)]support.[/url]

Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $$ AB+CD>\max(AM+DM,BN+CN)? $$ [i](Folklore)[/i]

Find, with proof, all ordered pairs of positive integers $(a, b)$ with the following property: there exist positive integers $r$, $s$, and $t$ such that for all $n$ for which both sides are defined, [center]${{n\choose{a}}\choose{b}}=r {{n+s}\choose{t}}$ .[/center]

Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$

In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$?

Each of $N$ people chooses a random integer number between $1$ and $19$ (including $1$ and $19$, and not necessarily with the same distribution). The random numbers chosen by the people are independent from each other, and it is true that each person chooses each of the $19$ numbers with probability at most $99\%$. They add up the $N$ chosen numbers, and take the remainder of the sum divided by $19$. Prove that the distribution of the result tends to the uniform distribution exponentially, i.e. there exists a number $0<c<1$ such that the mod $19$ remainder of the sum of the $N$ chosen numbers equals each of the mod $19$ remainders with probability between $\frac{1}{19}-c^{N}$ and $\frac{1}{19}+c^{N}$.

Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]