We say that a positive integer $n$ is [i]divable [/i] if there exist positive integers $1 < a < b < n$ such that, if the base-$a$ representation of $n$ is $\sum_{i=0}^{k_1} a_ia^i$ , and the base-$b$ representation of $n$ is $\sum_{i=0}^{k_2} b_ib^i$ , then for all positive integers $c > b$, we have that $\sum_{i=0}^{k_2} b_ic^i$ divides $\sum_{i=0}^{k_1} a_ic^i$. Find the number of non-divable $n$ such that $1 \le n \le 100$.

If the reals $a,b,c,d,e,f,g,h,i$ satisfy $$\begin{cases}ab+bc+ca=3\\de+ef+fd=3\\gh+hi+ig=3\\ad+dg+ga=3\\be+eh+hb=3\end{cases}$$show that $cf+fi+ic=3$ holds as well.

Let $ABC$ be a triangle with $\angle C=90^\circ$, and $A_0$, $B_0$, $C_0$ be the mid-points of sides $BC$, $CA$, $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$. Find the angle $C_0C_1C_2$.

Let $\mathcal{L}$ be the set of all lines in the plane and let $\mathcal{P}$ be the set of all points in the plane. Determine whether there exists a function $g : \mathcal{L} \to \mathcal{P}$ such that for any two distinct non-parallel lines $\ell_1, \ell_2 \in \mathcal{L}$, we have $(a)$ $g(\ell_1) \neq g(\ell_2)$, and $(b)$ if $\ell_3$ is the line passing through $g(\ell_1)$ and $g(\ell_2)$, then $g(\ell_3)$ is the intersection of $\ell_1$ and $\ell_2$.

Point $A$ lies inside an angle with vertex $M$. A ray issuing from point $A$ is reflected in one side of the angle at point $B$, then in the other side at point $C$ and then returns back to point $A$ (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle $\triangle BCM$ lies on line $AM$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $ABC$, $AB<AC$ be an acute triangle inscribed in circle $\Gamma$ with center $O$. The altitude from $A$ to $BC$ intersects $\Gamma$ again at $A_1$. $OA_1$ intersects $BC$ at $A_2$ Similarly define $B_1$, $B_2$, $C_1$, and $C_2$. Then $B_2C_2=2\sqrt{2}$. If $B_2C_2$ intersects $AA_2$ at $X$ and $BC$ at $Y$, then $XB_2=2$ and $YB_2=k$. Find $k^2$. [i]2017 CCA Math Bonanza Individual Round #15[/i]

Can the numbers $1,2,...,121$ be written in the cells of an $11\times 11$ board in such a way that any two consecutive numbers are in adjacent cells (sharing a side), and all perfect squares are in the same column?

For \(u,v \in\mathbb{R}^4\), let \(<u,v>\) denote the usual dot product. Define a [i]vector field[/i] to be a map \(\omega:\mathbb{R}\to\mathbb{R}\) such that \(<\omega(z),z>=0,\ \forall z\in\mathbb{R}^4.\) Find a maximal collection of vector fields \(\left\{\omega_1,...,\omega_k\right\}\) such that the map \(\Omega\) sending \(z\) to \(\lambda_1\omega_1(z)+\cdots+\lambda_k \omega_k(z)\), with \(\lambda_1,\ldots,\lambda_k\in\mathbb{R}\), is nonzero on \(\mathbb{R}^4\backslash\{0\}\) unless \(\lambda_1=\cdots=\lambda_k=0\)

Let $A$ be the area of the largest semicircle that can be inscribed in a quarter-circle of radius $ 1$. Compute$ \frac{120A}{\pi}$. .

A number $n$ is $\it{bad}$ if there exists some integer $c$ for which $x^x \equiv c \pmod n$ has no integer solutions for $x$. Find the number of bad integers between 2 and 42 inclusive.

Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.

Let $ABCD$ be a convex quadrilateral with $AB$ parallel to $CD$. Let $P$ and $Q$ be the midpoints of $AC$ and $BD$, respectively. Prove that if $\angle ABP=\angle CBD$, then $\angle BCQ=\angle ACD$.

Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property: $$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$

Let $k$ be a positive integer, and let $a$ be an integer such that $a-2$ is a multiple of $7$ and $a^6-1$ is a multiple of $7^k$. Prove that $(a + 1)^6-1$ is also a multiple of $7^k$.

For positive reals $a_1, a_2, ..., a_5$ such that $a^2_1+a^2_2+...+a^2_5=2$, consider five squares with sides $a_1, a_2, ..., a_5$ respectively. Show that these squares can be placed inside (including boundaries) a square with side length of $2$ so that the square themselves do not overlap each other.