At most how many positive integers less than $51$ are there such that no one is triple of another one? $ \textbf{(A)}\ 17 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 38 \qquad\textbf{(D)}\ 39 \qquad\textbf{(E)}\ \text{None of the preceding} $

Find all positive integers $n$ such that $$\lfloor \sqrt{n} \rfloor + \left\lfloor \frac{n}{\lfloor \sqrt{n} \rfloor} \right \rfloor> 2\sqrt{n}.$$ If $k$ is a real number, then $\lfloor k \rfloor$ means the floor of $k$, this is the greatest integer less than or equal to $k$.

Let $n$ be a positive integer. Prove that $$\gcd(\underbrace{11\dots 1}_{n \text{times}},n)\mid 1+10^k+10^{2k}+\dots+10^{(n-1)k}$$ for all positive integer $k$.

Let the incircle of triangle $ABC$ meet the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, and let the $A$-excircle meet the lines $BC$, $CA$, $AB$ at $P$, $Q$, $R$. Let the line passing through $A$ and perpendicular to $BC$ meet the lines $EF$, $QR$ at $K$, $L$. Let the intersection of $LD$ and $EF$ be $S$, and the intersection of $KP$ and $QR$ be $T$. Prove that $A$, $S$, $T$ are collinear.

Given is a triangle $ABC$ with altitude $AH$ and median $AM$. The line $OH$ meets $AM$ at $D$. Let $AB \cap CD=E, AC \cap BD=F$. If $EH$ and $FH$ meet $(ABC)$ at $X, Y$, prove that $BY, CX, AH$ are concurrent.

Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$'s.

It is known that a circle can be inscribed in a trapezium $ABCD$. Prove that the two circles, constructed on its oblique sides as diameters, touch each other. (D. Fomin, Leningrad)

In this question we re-define the operations addition and multiplication as follows: $a + b$ is defined as the minimum of $a$ and $b$, while $a * b$ is defined to be the sum of $a$ and $b$. For example, $3+4 = 3$, $3*4 = 7$, and $$3*4^2+5*4+7 = \min(\text{3 plus 4 plus 4}, \text{5 plus 4}, 7) = \min(11, 9, 7) = 7.$$ Let $a, b, c$ be real numbers. Characterize, in terms of $a, b, c$, what the graph of $y = ax^2+bx+c$ looks like.

A random rectangle (not necessarily a square) with positive integer dimensions is selected from the $2\times4$ grid below. The probability that the selected rectangle contains only white squares can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [asy] fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,blue); fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,blue); draw((0,0)--(4,0),black); draw((0,0)--(0,2),black); draw((4,0)--(4,2),black); draw((4,2)--(0,2),black); draw((0,1)--(4,1),black); draw((1,0)--(1,2),black); draw((2,0)--(2,2),black); draw((3,0)--(3,2),black); [/asy]

Three circles $k_1, k_2$, and $k_3$ intersect in point $O$. Let $A, B$, and $C$ be the second intersection points (other than $O$) of $k_2$ and $k_3, k_1$ and $k_3$, and $k_1$ and $k_2$, respectively. Assume that $O$ lies inside of the triangle $ABC$. Let lines $AO,BO$, and $CO$ intersect circles $k_1, k_2$, and $k_3$ for a second time at points $A', B'$, and $C'$, respectively. If $|XY|$ denotes the length of segment $XY$, prove that $\frac{|AO|}{|AA'|}+\frac{|BO|}{|BB'|}+\frac{|CO|}{|CC'|}= 1$

A tourist is learning an incorrect way to sort a permutation $(p_1, \dots, p_n)$ of the integers $(1, \dots, n)$. We define a [i]fix[/i] on two adjacent elements $p_i$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_i>p_{i+1}$, and does nothing otherwise. The tourist performs $n-1$ rounds of fixes, numbered $a=1, 2, \dots, n-1$. In round $a$ of fixes, the tourist fixes $p_a$ and $p_{a+1}$, then $p_{a+1}$ and $p_{a+2}$, and so on, up to $p_{n-1}$ and $p_n$. In this process, there are $(n-1)+(n-2)+\dots+1 = \tfrac{n(n-1)}{2}$ total fixes performed. How many permutations of $(1, \dots, 2018)$ can the tourist start with to obtain $(1, \dots, 2018)$ after performing these steps?

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Consider the configurations of integers $a_{1,1}$ $a_{2,1} \quad a_{2,2}$ $a_{3,1} \quad a_{3,2} \quad a_{3,3}$ $\dots \quad \dots \quad \dots$ $a_{2017,1} \quad a_{2017,2} \quad a_{2017,3} \quad \dots \quad a_{2017,2017}$ Where $a_{i,j} = a_{i+1,j} + a_{i+1,j+1}$ for all $i,j$ such that $1 \leq j \leq i \leq 2016$. Determine the maximum amount of odd integers that such configuration can contain.

A plane $p$ passes through a vertex of a cube so that the three edges at the vertex make equal angles with $p$. Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto $p$. There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane $p$.

Find all quadruples of real numbers $(a,b,c,d)$ that satisfy the system of equations: \begin{align*} a+4b+8c+4d&=53\\ 3a^2+4b^2+12c^2+2d^2&=159\\ 9a^3+4b^3+18c^3+d^3&=477. \end{align*}

Given that nonzero reals $a,b,c,d$ satisfy $a^b=c^d$ and $\frac{a}{2c}=\frac{b}{d}=2$, compute $\frac{1}{c}$. [i]2021 CCA Math Bonanza Lightning Round #2.2[/i]

For any positive integer $n$ consider all representations $n = a_1 + \cdots+ a_k$, where $a_1 > a_2 > \cdots > a_k > 0$ are integers such that for all $i \in \{1, 2, \cdots , k - 1\}$, the number $a_i$ is divisible by $a_{i+1}$. Find the longest such representation of the number $1992.$

Let $ {{a} _ {1}}, {{a} _ {2}}, \ldots, {{a} _ {n}} $ be permutation of numbers $ 1,2, \ldots, n $, where $ n \geq 2 $. Find the maximum value of the sum $$ S (n) = | {{a} _ {1}} - {{a} _ {2}} | + | {{a} _ {2}} - {{a} _ {3}} | + \cdots + | {{a} _ {n-1}} - {{a} _ {n}} |. $$

Let $p$ be a fixed odd prime. Find the minimum positive value of $E_{p}(x,y) = \sqrt{2p}-\sqrt{x}-\sqrt{y}$ where $x,y \in \mathbb{Z}_{+}$.

Let $ 0\neq\varrho\in\text{Hom}\left( \mathbb{Z}_4,\mathbb{Z}_2\right) ,$ $ \text{id}\neq\iota\in\text{Aut}\left( \mathbb{Z}_4\right) ,$ $ G:=\left\{ (x,y)\in\mathbb{Z}_4^2\big|x-y\in\ker\varrho\right\} , $ and $ \rho_1,\rho_2, $ the canonic projections of $ G $ into $ \mathbb{Z}_4. $ Prove that there exists an unique $ \nu\in\text{Hom}\left( \mathbb{Z}_4,G\right) $ such that $ \rho_1\circ\nu=\text{id} $ and $ \rho_2\circ\nu =\iota . $ Determine numerically this morphism.

Let $U$ be the center of the circumcircle of the acute-angled triangle $ABC$. Let $M_A, M_B$ and $M_C$ be the circumcenters of triangles $UBC, UAC$ and $UAB$ respecrively. For which triangles $ABC$ is the triangle $M_AM_BM_C$ similar to the starting triangle (with a suitable order of the vertices)?

(Eisentein's Criterion) Let $f(x)=a_{n}x^{n} +\cdots +a_{1}x+a_{0}$ be a nonconstant polynomial with integer coefficients. If there is a prime $p$ such that $p$ divides each of $a_{0}$, $a_{1}$, $\cdots$,$a_{n-1}$ but $p$ does not divide $a_{n}$ and $p^2$ does not divide $a_{0}$, then $f(x)$ is irreducible in $\mathbb{Q}[x]$.

In triangle $ABC$ we have $|AB| \ne |AC|$. The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$, respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$.

Let $ \triangle{XOY}$ be a right-angled triangle with $ m\angle{XOY}\equal{}90^\circ$. Let $ M$ and $ N$ be the midpoints of legs $ OX$ and $ OY$, respectively. Given that $ XN\equal{}19$ and $ YM\equal{}22$, find $ XY$. $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

A [i]unitary [/i] divisor d of a number $n$ is a divisor $n$ that has the property $\gcd (d, n/d) = 1$. If $n = 1620$, what is the sum of all of the unitary divisors of $d$?