The following construction is used for training astronauts: A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a way that the touching point of the circles $C_2$ and $C_3$ remains at maximum distance from the touching point of the circles $C_1$ and $C_2$ at all times. How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_3$ while the circle $C_2$ completes one full lap around the inside of circle $C_1$?

The number $60$ is written on a blackboard. In every move, Andris wipes the numbers on the board one by one, and writes all its divisors in its place (including itself). After $10$ such moves, how many times will $1$ appear on the board?

For a positive integer $n$, let $d\left(n\right)$ be the number of positive divisors of $n$ (for example $d\left(39\right)=4$). Estimate the average value that $d\left(n\right)$ takes on as $n$ ranges from $1$ to $2019$. An estimate of $E$ earns $2^{1-\left|A-E\right|}$ points, where $A$ is the actual answer. [i]2019 CCA Math Bonanza Lightning Round #5.3[/i]

How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.) [i]Based on a proposal by Yannick Yao[/i]

Let real numbers $a_1, a_2, \ldots, a_n$ satisfy $0 < a_i \leq a, \ i = 1, 2, \ldots, n$. Prove that (i) If $n = 4$, then \[\frac 1a \sum_{i=1}^4 a_i - \frac{a_1a_2 + a_2a_3 + a_3 a_4 + a_4 a_1}{a^2} \leq 2.\] (ii) If $n = 6$, then \[\frac 1a \sum_{i=1}^6 a_i - \frac{a_1a_2 + a_2a_3 + \cdots + a_5 a_6 + a_6 a_1}{a^2} \leq 3.\]

Real numbers $a,b,c$ with $a\neq b$ verify $$a^2(b+c)=b^2(c+a)=2023.$$ Find the numerical value of $E=c^2(a+b)$.

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

$P,Q$ are two fixed points on a circle centered at $O$, and $M$ is an interior point of the circle that differs from $O$. $M,P,Q,O$ are concyclic. Prove that the bisector of $\angle PMQ$ is perpendicular to line $OM$.

Let $p_1$, $p_2$ be two odd prime numbers and $\alpha $, $n$ be positive integers with $\alpha >1$, $n>1$. Prove that if the equation $\left (\frac{p_2 -1}{2} \right )^{p_1} + \left (\frac{p_2 +1}{2} \right )^{p_1} = \alpha^n$ does not have integer solutions for both $p_1 =p_2$ and $p_1 \neq p_2$.

Three parallel lines $\ell_a, \ell_b, \ell_c$ pass through the vertices of triangle $ABC$. A line $a$ is the reflection of altitude $AH_a$ about $\ell_a$. Lines $b, c$ are defined similarly. Prove that $a, b, c$ are concurrent.

Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.

Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N.$ For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double?

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.

A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window? $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 36$

A quadrilateral ABCD is inscribed into a circle ω with center O. The circumcircle of the triangle AOC intersects the lines AB, BC, CD and DA the second time at the points M, N, K and L respectively. Prove that the lines MN, KL and the tangents to ω at the points A и C all touch the same circle.

In the isosceles triangle $ABC$, with $AB=AC$, the angle bisector of $\angle B$ meets the side $AC$ at $B'$. Suppose that $BB'+B'A=BC$. Find the angles of the triangle $ABC$. [i]Dan Nedeianu[/i]

Find all the natural numbers $m,n$ such that $1+5 \cdot 2^m=n^2$.

Let $a_{1}$, $a_{2}$, …, $a_{6}$; $b_{1}$, $b_{2}$, …, $b_{6}$ and $c_{1}$, $c_{2}$, …, $c_{6}$ are all permutations of $1$, $2$, …, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$.

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$. Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$. [i]Proposed by Anant Mudgal, India[/i]

Consider the square $ABCD$ and $E \in (AB)$. The diagonal $AC$ intersects the segment $[DE]$ at point $P$. The perpendicular taken from point $P$ on $DE$ intersects the side $BC$ at point $F$. Prove that $EF = AE + FC$.

Let $f(n)$ be the product of all factors of $n$. Find all natural numbers $n$ such that $f(n)$ is not a perfect power of $n$.

The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$. Determine the value of $P(1)/P(-1)$.

If there is only $1$ complex solution to the equation $8x^3 + 12x^2 + kx + 1 = 0$, what is $k$?

The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has $n$ aluminum coins and $n$ bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k \leq 2n$, Gilberty repeatedly performs the following operation: he identifies the longest chain containing the $k^{th}$ coin from the left and moves all coins in that chain to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the ordering $AABBBABA$ would be $AABBBABA \to BBBAAABA \to AAABBBBA \to BBBBAAAA \to ...$ Find all pairs $(n,k)$ with $1 \leq k \leq 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.

i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is: Find all positive integers m for which for all $\alpha,\beta \in \mathbb{Z}-\{0\}$ \[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]