Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]G.16[/b] A number $k$ is the product of exactly three distinct primes (in other words, it is of the form $pqr$, where $p, q, r$ are distinct primes). If the average of its factors is $66$, find $k$. [b]G.17[/b] Find the number of lattice points contained on or within the graph of $\frac{x^2}{3} +\frac{y^2}{2}= 12$. Lattice points are coordinate points $(x, y)$ where $x$ and $y$ are integers. [b]G.18 / C.23[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct. [b]G.19[/b] Cindy has a cone with height $15$ inches and diameter $16$ inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint? [b]G.20 / C.25[/b] An even positive integer $n$ has an odd factorization if the largest odd divisor of $n$ is also the smallest odd divisor of n greater than 1. Compute the number of even integers $n$ less than $50$ with an odd factorization. [u] Set 5[/u] [b]G.21[/b] In the magical tree of numbers, $n$ is directly connected to $2n$ and $2n + 1$ for all nonnegative integers n. A frog on the magical tree of numbers can move from a number $n$ to a number connected to it in $1$ hop. What is the least number of hops that the frog can take to move from $1000$ to $2018$? [b]G.22[/b] Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for $10$ days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy $1024$ dollars, but then he learns the catch - after $10$ days, the amount of money he has must be a multiple of $11$ or he loses all his money. What is the largest amount of money Stan can have after the $10$ days are up? [b]G.23[/b] Let $\Gamma_1$ be a circle with diameter $2$ and center $O_1$ and let $\Gamma_2$ be a congruent circle centered at a point $O_2 \in \Gamma_1$. Suppose $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. Let $\Omega$ be a circle centered at $A$ passing through $B$. Let $P$ be the intersection of $\Omega$ and $\Gamma_1$ other than $B$ and let $Q$ be the intersection of $\Omega$ and ray $\overrightarrow{AO_1}$. Define $R$ to be the intersection of $PQ$ with $\Gamma_1$. Compute the length of $O_2R$. [b]G.24[/b] $8$ people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible? [b]G.25[/b] Let $S$ be the set of points $(x, y)$ such that $y = x^3 - 5x$ and $x = y^3 - 5y$. There exist four points in $S$ that are the vertices of a rectangle. Find the area of this rectangle. PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$

Let $n> 1$ be an odd integer. On a square surface have been placed $n^2 - 1$ white slabs and a black slab on the center. Two workers $A$ and $B$ take turns removing them, betting that whoever removes black will lose. First $A$ picks a slab; if it has row number $i \ge (n + 1) / 2$, then it will remove all tiles from rows with number greater than or equal to$ i$, while if $i <(n + 1) / 2$, then it will remove all tiles from the rows with lesser number or equal to $i$. Proceed in a similar way with columns. Then $B$ chooses one of the remaining tiles and repeats the process. Determine who has a winning strategy and describe it. Note: Row and column numbering is ascending from top to bottom and from left to right.

Suppose that $A\in\mathcal M_{n}(\mathbb R)$ with $\text{Rank}(A)=k$. Prove that $A$ is sum of $k$ matrices $X_{1},\dots,X_{k}$ with $\text{Rank}(X_{i})=1$.

Consider the second degree polynomial $x^2+ax+b$ with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant, $a^2-4b$ be greater than or equal to zero. Note that the discriminant is also a polynomial with variables $a$ and $b$. Prove that the same story is not true for polynomials of degree $4$: Prove that there does not exist a $4$ variable polynomial $P(a,b,c,d)$ such that: The fourth degree polynomial $x^4+ax^3+bx^2+cx+d$ can be written as the product of four $1$st degree polynomials if and only if $P(a,b,c,d)\ge 0$. (All the coefficients are real numbers.) [i]Proposed by Sahand Seifnashri[/i]

$ABCD$ is any convex quadrilateral. Construct a new quadrilateral as follows. Take $A'$ so that $A$ is the midpoint of $DA'$; similarly, $B'$ so that $B$ is the midpoint of $AB'$; $C'$ so that $C$ is the midpoint of $BC'$; and $D'$ so that $D$ is the midpoint of $CD'$. Show that the area of $A'B'C'D'$ is five times the area of $ABCD$.

For all $x\in\left(0,\frac\pi4\right)$ prove $$\frac{(\sin^2x)^{\sin^2x}+(\tan^2x)^{\tan^2x}}{(\sin^2x)^{\tan^2x}+(\tan^2x)^{\sin^2x}}<\frac{\sin x}{4\sin x-3x}$$ [i]Proposed by Pirkulyiev Rovsen[/i]

It is possible to write the numbers $111$, $112$, $121$, $122$, $211$, $212$, $221$ and $222$ at the vertices of a cube, so that the numbers written in adjacent vertices match at most in one digit?

Find the remainder when $$\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor$$ is divided by $10^4$. [i]Proposed by Vijay Srinivasan[/i]

Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $$2 * (5 * x)=1?$$ $\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14$

Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet? $ \textbf{(A)}\ 10: 00 \qquad \textbf{(B)}\ 10: 15 \qquad \textbf{(C)}\ 10: 30 \qquad \textbf{(D)}\ 11: 00 \qquad \textbf{(E)}\ 11: 30$

A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid? $\textbf{(A) }18\qquad\textbf{(B) }162\qquad\textbf{(C) }36\sqrt{21}\qquad\textbf{(D) }18\sqrt{138}\qquad\textbf{(E) }54\sqrt{21}$

If $ \log(xy^3)\equal{}1$ and $ \log(x^2y)\equal{}1$, what is $ \log(xy)$? $ \textbf{(A)}\ \minus{}\!\frac{1}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{3}{5} \qquad \textbf{(E)}\ 1$

The area of a triangle is $S$ and the sum of the lengths of its sides is $L$. Prove that $36S \leq L^2\sqrt 3$ and give a necessary and sufficient condition for equality.

Find all positive integers $n$ that are equal to the sum of digits of $n^2$.

Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says: "Bret is next to Carl." "Abby is between Bret and Carl." However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2? $\text{(A)}\ \text{Abby} \qquad \text{(B)}\ \text{Bret} \qquad \text{(C)}\ \text{Carl} \qquad \text{(D)}\ \text{Dana} \qquad \text{(E)}\ \text{There is not enough information to be sure.}$

Given a triangle $ABC$, in which $\angle B> 90^o$. Perpendicular bisector of the side $AB$ intersects the side $AC$ at the point $M$, and the perpendicular bisector of the side $AC$ intersects the extension of the side $AB$ beyond the vertex $B$ at point $N$. It is known that the segments $MN$ and $BC$ are equal and intersect at right angles. Find the values ​​of all angles of triangle $ABC$.

Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.

In the logarithm table below, there are two mistakes. Correct them. \begin{tabular}{|c|c|} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $\lg0.021$&$2a+b+c-3$ \\ \hline $\lg0.27$&$6a-3b-2$\\ \hline $\lg1.5$&$3a-b+c$\\ \hline $\lg2.8$&$1-2a+2b-c$\\ \hline $\lg3$&$2a-b$\\ \hline $\lg5$&$a+c$\\ \hline $\lg6$&$1+a-b-c$\\ \hline $\lg7$&$2(a+c)$\\ \hline $\lg8$&$3-3a-3c$\\ \hline $\lg9$&$4a-2b$\\ \hline $\lg14$&$1-a+2b$\\ \hline \end{tabular}

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]

Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$. Find the number of Juan.

Given nonzero real numbers a,b,c with $a+b+c=a^2+b^2+c^2=a^3+b^3+c^3$. ($*$) a) Find $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)$ b) Do there exist pairwise different nonzero $a,b,c$ satisfying ($*$)? D. Bazylev