Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

Let $x$ be a real number with the following property: for each positive integer $q$, there exists an integer $p$, such that \[\left|x-\frac{p}{q} \right|<\frac{1}{3q}. \] Prove that $x$ is an integer.

Let $a_n$ be a sequence of positive numbers such that: i) $\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}$, for every $n\in\mathbb{N}^{\star}$ ii) $\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1$, for every $ k,n\in\mathbb{N}^{\star}$ with $|k-n|\neq 1$. (a) Prove that $(a_n)$ is a geometric progression. (n) Prove that exists $t>0$, such that $\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t$

For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96)=1+2+3+6+9=21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a)=n$, where $f^k(a)$ denotes $f$ applied $k$ times to $a$. [i]2021 CCA Math Bonanza Lightning Round #4.3[/i]

Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.

The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.

Let $a_1, a_2, \ldots, a_n$ be real numbers such that $\sum_{i=1}^n a_i = n$, $\sum_{i = 1}^n a_i^2 = 2n$, $\sum_{i=1}^n a_i^3 = 3n$. (i) Find the largest constant $C$, such that for all $n \geqslant 4$, \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C. \] (ii) Prove that there exists a positive constant $C_2$, such that \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C + C_2 n^{-\frac 32}, \]where $C$ is the constant determined in (i).

Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$ with $a+b+c=1 \}$ and $f: M\to R$ given as $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$ Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.

The math problem book contains $300$ problems. The teacher has cards with numbers. She pins these cards to a special stand and indicates the numbers of four problems that need to be solved during the lesson. What is the smallest number of cards that a teacher can use in order to be able to indicate the numbers of any four problems from the problem book?

A $13$-foot tall extraterrestrial is standing on a very small spherical planet with radius $156$ feet. It sees an ant crawling along the horizon. If the ant circles the extraterrestrial once, always staying on the horizon, how far will it travel (in feet)?

A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.

Let $n\geq 2$ be an integer. The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called [i]quadratic[/i] if $a_ia_{i +1} + 1$ is a perfect square for all $1\leq i \leq n-1.$ The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called [i]cubic[/i] if $a_ia_{i + 1} + 1$ is a perfect cube for all $1\leq i \leq n - 1.$ a) Prove that for infinitely many values of $n$ is there at least one quadratic permutation of the numbers $1, 2,...,n.$ b) Prove that for no value of $n$ is there a cubic permutation of the numbers $1, 2,..., n.$

Label the squares of a $1991 \times 1992$ rectangle $(m, n)$ with $1 \leq m \leq 1991$ and $1 \leq n \leq 1992$. We wish to color all the squares red. The first move is to color red the squares $(m, n), (m+1, n+1), (m+2, n+1)$for some $m < 1990, n < 1992$. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?

Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]

Circles $w_1$ and $w_2$ intersect at points $P$ and $Q$ and touch a circle $w$ with center at point $O$ internally at points $A$ and $B$, respectively. It is known that the points $A,B$ and $Q$ lie on one line. Prove that the point $O$ lies on the external bisector $\angle APB$. (Nazar Serdyuk)

Let $a_1, a_2,...$ be an infinite sequence of positive integers such that for any $k,\ell\in \mathbb{Z_+}$, $a_{k+\ell}$ is divisible by $\gcd(a_k,a_\ell)$. Prove that for any integers $1\leqslant k\leqslant n$, $a_na_{n-1}\dots a_{n-k+1}$ is divisible by $a_ka_{k-1}\dots a_1$.

In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is $X$, in centimeters? [asy] pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0); B=(7,0); C=(7,4); D=(8,4); E=(8,5); F=(10,5); G=(10,7); H=(7,7); I=(7,8); J=(5,8); K=(5,7); L=(4,7); M=(4,6); N=(0,6); O=(0,5); P=(2,5); Q=(2,3); R=(4,3); draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle); label("$X$",(3.4,1.5)); label("6",(7.6,1.5)); label("1",(7.6,3.5)); label("1",(8.4,4.6)); label("2",(9.4,4.6)); label("2",(10.4,6)); label("3",(8.4,7.4)); label("1",(7.5,7.8)); label("2",(6,8.5)); label("1",(4.7,7.8)); label("1",(4.3,7.5)); label("1",(3.5,6.5)); label("4",(1.8,6.5)); label("1",(-0.5,5.5)); label("2",(0.8,4.5)); label("2",(1.5,3.8)); label("2",(2.8,2.6)); [/asy] $\textbf{(A)}\hspace{.05in}1 \qquad \textbf{(B)}\hspace{.05in}2 \qquad \textbf{(C)}\hspace{.05in}3 \qquad \textbf{(D)}\hspace{.05in}4 \qquad \textbf{(E)}\hspace{.05in}5 $

We have $10$ points on a line $A_1,A_2\ldots A_{10}$ in that order. Initially there are $n$ chips on point $A_1$. Now we are allowed to perform two types of moves. Take two chips on $A_i$, remove them and place one chip on $A_{i+1}$, or take two chips on $A_{i+1}$, remove them, and place a chip on $A_{i+2}$ and $A_i$ . Find the minimum possible value of $n$ such that it is possible to get a chip on $A_{10}$ through a sequence of moves.

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

We have a large supply of black, white, red and green hats. And we want to give $8$ of these hats to $8$ students that are sitting around a round table. Find the number of ways of doing that in each of these cases (assuming for the purposes of this problem that students will notchange their places, and that hats of the same color are identical) a) Each hat to be used must be either red or green. b) Exactly two hats of each color are to be used c) Exactly two hats of each color are to be used, and every two hats of the same color are to be given to two adjacent students. d) Exactly two hats of each color are to be used, and no two hats of the same color are to be given to two adjacent students. e) There are no restrictions on the number of hats of each color that are to be used, but no two hats of the same color are to be given to two adjacent students.

The value of $ 10^{\log_{10}7}$ is: $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ \log_{10} 7 \qquad\textbf{(E)}\ \log_7 10$

Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.

We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$, and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$. If the line $RS$ meets the extension of the diameter at $Z$, prove that $ZC$ is tangent to the semicircle.

Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$?