Prove that there is a positive integer $n$ such that the decimal representation of $7^n$ contains a block of at least $m$ consecutive zeros, where $m$ is any given positive integer.

Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that \[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]

Three children wanted to make a table-game. For that purpose they wished to enumerate the $mn$ squares of an $m \times n$ game-board by the numbers $1, ... ,mn$ in such way that the numbers $1$ and $mn$ lie in the corners of the board and the squares with successive numbers have a common edge. The children agreed to place the initial square (with number $1$) in one of the corners but each child wanted to have the final square (with number $mn$ ) in different corner. For which numbers $m$ and $n$ is it possible to satisfy the wish of any of the children?

Show that for nonnegative integers $m$ and $n$, $\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$ $=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.

For every positive integer $n$, $P(n)$ is defined as follows: For each prime divisor $p$ of $n$ is considered the largest integer $k$ such that $p^k\le n$ and all the $p^k$ are added. For example, for $n=100=2^2 \cdot 5^2$, as $2^6<100<2^7$ and $5^2<100<5^3$, it turns out that $P(100)=2^6+5^2=89$ Prove that there are infinitely many positive integers $n$ such that $P(n)>n$..

Find all integer/s $n$ such that $\displaystyle{\frac{5^n-1}{3}}$ is a prime or a perfect square of an integer. [i]Proposed by Prajit Adhikari, Nepal[/i]

Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.

by using the formula $\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2}$ calculate values of $\zeta(2k)$ on terms of bernoli numbers and powers of $\pi$.

Let $n$ be a positive integer. If the number $1998$ is written in base $n$, a three-digit number with the sum of digits equal to $24$ is obtained. Find all possible values of $n$.

Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties: $\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order. $\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)

Let $ABC$ be an acute triangle, $\omega$ its circumcircle and $O$ its circumcenter. The altitude from $A$ intersects $\omega$ in a point $D \ne A$ and the segment $AC$ intersects the circumcircle of $OCD$ in a point $E \ne C$. Finally, let $M$ be the midpoint of $BE$. Show that $DE$ is parallel to $OM$.

The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made? [asy] unitsize(1 cm); real unitwidth, dayheight, barheight; int i; unitwidth = 0.5; dayheight = 1; barheight = 0.3; draw((unitwidth,0)--(unitwidth,5*dayheight),gray(0.7)); draw((2*unitwidth,0)--(2*unitwidth,5*dayheight),gray(0.7)); draw((3*unitwidth,0)--(3*unitwidth,5*dayheight),gray(0.7)); draw((4*unitwidth,0)--(4*unitwidth,5*dayheight),gray(0.7)); draw((5*unitwidth,0)--(5*unitwidth,5*dayheight),gray(0.7)); draw((6*unitwidth,0)--(6*unitwidth,5*dayheight),gray(0.7)); draw((7*unitwidth,0)--(7*unitwidth,5*dayheight),gray(0.7)); fill((0,1/2*dayheight - 1/2*barheight)--(8*unitwidth,1/2*dayheight - 1/2*barheight)--(8*unitwidth,1/2*dayheight + 1/2*barheight)--(0,1/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); fill((0,5/2*dayheight - 1/2*barheight)--(8*unitwidth,5/2*dayheight - 1/2*barheight)--(8*unitwidth,5/2*dayheight + 1/2*barheight)--(0,5/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); draw((8*unitwidth,0)--(8*unitwidth,5*dayheight),gray(0.7)); draw((9*unitwidth,0)--(9*unitwidth,5*dayheight),gray(0.7)); fill((0,9/2*dayheight - 1/2*barheight)--(10*unitwidth,9/2*dayheight - 1/2*barheight)--(10*unitwidth,9/2*dayheight + 1/2*barheight)--(0,9/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); draw((10*unitwidth,0)--(10*unitwidth,5*dayheight),gray(0.7)); fill((0,3/2*dayheight - 1/2*barheight)--(11*unitwidth,3/2*dayheight - 1/2*barheight)--(11*unitwidth,3/2*dayheight + 1/2*barheight)--(0,3/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); draw((11*unitwidth,0)--(11*unitwidth,5*dayheight),gray(0.7)); draw((12*unitwidth,0)--(12*unitwidth,5*dayheight),gray(0.7)); fill((0,7/2*dayheight - 1/2*barheight)--(13*unitwidth,7/2*dayheight - 1/2*barheight)--(13*unitwidth,7/2*dayheight + 1/2*barheight)--(0,7/2*dayheight + 1/2*barheight)--cycle,gray(0.5)); draw((0*unitwidth,0)--(0*unitwidth,5*dayheight),gray(0.7)); draw((13*unitwidth,0)--(13*unitwidth,5*dayheight),gray(0.7)); draw((14*unitwidth,0)--(14*unitwidth,5*dayheight),gray(0.7)); label("$0$", (0,5*dayheight), N); label("$4$", (2*unitwidth,5*dayheight), N); label("$8$", (4*unitwidth,5*dayheight), N); label("$12$", (6*unitwidth,5*dayheight), N); label("$16$", (8*unitwidth,5*dayheight), N); label("$20$", (10*unitwidth,5*dayheight), N); label("$24$", (12*unitwidth,5*dayheight), N); label("$28$", (14*unitwidth,5*dayheight), N); label("Number of students at soccer practice", (7*unitwidth,6*dayheight)); label("Monday", (-0.5*unitwidth,9/2*dayheight), W); label("Tuesday", (-0.5*unitwidth,7/2*dayheight), W); label("Wednesday", (-0.5*unitwidth,5/2*dayheight), W); label("Thursday", (-0.5*unitwidth,3/2*dayheight), W); label("Friday", (-0.5*unitwidth,1/2*dayheight), W); [/asy] $\textbf{(A) } \text{The mean increases by 1 and the median does not change.}$ $\textbf{(B) } \text{The mean increases by 1 and the median increases by 1.}$ $\textbf{(C) } \text{The mean increases by 1 and the median increases by 5.}$ $\textbf{(D) } \text{The mean increases by 5 and the median increases by 1.}$ $\textbf{(E) } \text{The mean increases by 5 and the median increases by 5.}$

Let $p, q$ be integers and $p, q > 1$ , $gcd(p, \,6q)=1$. Prove that:$$\sum_{k=1}^{q-1}\left \lfloor \frac{pk}{q}\right\rfloor^2 \equiv 2p \sum_{k=1}^{q-1}k\left\lfloor \frac{pk}{q} \right\rfloor (mod \, q-1)$$

We call a natural number venerable if the sum of all its divisors, including $1$, but not including the number itself, is $1$ less than this number. Find all the venerable numbers, some exact degree of which is also venerable.

Three identical circles are packed into a unit square. Each of the three circles are tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\tfrac{1}{r})^2$?

For a positive integer $n$, denote $c_n=2017^n$. A function $f: \mathbb{N} \rightarrow \mathbb{R}$ satisfies the following two conditions. 1. For all positive integers $m, n$, $f(m+n) \le 2017 \cdot f(m) \cdot f(n+325)$. 2. For all positive integer $n$, we have $0<f(c_{n+1})<f(c_n)^{2017}$. Prove that there exists a sequence $a_1, a_2, \cdots $ which satisfies the following. For all $n, k$ which satisfies $a_k<n$, we have $f(n)^{c_k} < f(c_k)^n$.

Let $ P$ be a point in the interior of a convex $ n$-gon $ A_1 A_2 ... A_n$ $ (n \ge 3)$. Show that among the angles $ \beta _{ij}\equal{}\angle A_i P A_j,1 \le i \le n$, there are at least $ n\minus{}1$ angles satisfying $ 90^{\circ} \le \beta_{ij} \le 180^{\circ}$.

Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

Let $ABC$ be a triangle and let $X$ and $Y$ be points on the $A$-symmedian such that $AX = XB$ and $AY = YC$. Let $BX$ and $CY$ meet at $Z$. Let the $Z$-excircle of triangle $XYZ$ touch $ZX$ and $ZY$ at $E$ and $F$. Show that $A$, $E$, $F$ are collinear.

Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$? If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.

Divide a segment in halves using a right triangle. (With a right triangle one can draw straight lines and erect perpendiculars but cannot draw perpendiculars.)

Let a polynomial $f(x)$ be given with real coefficients and has degree greater or equal than 1. Show that for every real number $c > 0$, there exists a positive integer $n_0$ satisfying the following condition: if polynomial $P(x)$ of degree greater or equal than $n_0$ with real coefficients and has leading coefficient equal to 1 then the number of integers $x$ for which $|f(P(x))| \leq c$ is not greater than degree of $P(x)$.

[b]p1.[/b] Eric is playing Brawl Stars. If he starts playing at $11:10$ AM, and plays for $2$ hours total, then how many minutes past noon does he stop playing? [b]p2.[/b] James is making a mosaic. He takes an equilateral triangle and connects the midpoints of its sides. He then takes the center triangle formed by the midsegments and connects the midpoints of its sides. In total, how many equilateral triangles are in James’ mosaic? [b]p3.[/b] What is the greatest amount of intersections that $3$ circles and $3$ lines can have, given that they all lie on the same plane? [b]p4.[/b] In the faraway land of Arkesia, there are two types of currencies: Silvers and Gold. Each Silver is worth $7$ dollars while each Gold is worth $17$ dollars. In Daniel’s wallet, the total dollar value of the Silvers is $1$ more than that of the Golds. What is the smallest total dollar value of all of the Silvers and Golds in his wallet? [b]p5.[/b] A bishop is placed on a random square of a $8$-by-$8$ chessboard. On average, the bishop is able to move to $s$ other squares on the chessboard. Find $4s$. Note: A bishop is a chess piece that can move diagonally in any direction, as far as it wants. [b]p6.[/b] Andrew has a certain amount of coins. If he distributes them equally across his $9$ friends, he will have $7$ coins left. If he apportions his coins for each of his $15$ classmates, he will have $13$ coins to spare. If he splits the coins into $4$ boxes for safekeeping, he will have $2$ coins left over. What is the minimum number of coins Andrew could have? [b]p7.[/b] A regular polygon $P$ has three times as many sides as another regular polygon $Q$. The interior angle of $P$ is $16^o$ greater than the interior angle of $Q$. Compute how many more diagonals $P$ has compared to $Q$. [b]p8.[/b] In an certain airport, there are three ways to switch between the ground floor and second floor that are 30 meters apart: either stand on an escalator, run on an escalator, or climb the stairs. A family on vacation takes 65 seconds to climb up the stairs. A solo traveller late for their flight takes $25$ seconds to run upwards on the escalator. The amount of time (in seconds) it takes for someone to switch floors by standing on the escalator can be expressed as $\frac{u}{v}$ , where $u$ and $v$ are relatively prime. Find $u + v$. (Assume everyone has the same running speed, and the speed of running on an escalator is the sum of the speeds of riding the escalator and running on the stairs.) [b]p9.[/b] Avanish, being the studious child he is, is taking practice tests to improve his score. Avanish has a $60\%$ chance of passing a practice test. However, whenever Avanish passes a test, he becomes more confident and instead has a $70\%$ chance of passing his next immediate test. If Avanish takes $3$ practice tests in a row, the expected number of practice tests Avanish will pass can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime. Find $a + b$. [b]p10.[/b] Triangle $\vartriangle ABC$ has sides $AB = 51$, $BC = 119$, and $AC = 136$. Point $C$ is reflected over line $\overline{AB}$ to create point $C'$. Next, point $B$ is reflected over line $\overline{AC'}$ to create point $B'$. If $[B'C'C]$ can be expressed in the form of $a\sqrt{b}$, where $b$ is not divisible by any perfect square besides $1$, find $a + b$. [b]p11[/b]. Define the following infinite sequence $s$: $$s = \left\{\frac{1}{1},\frac{1}{1 + 3},\frac{1}{1 + 3 + 6}, ... ,\frac{1}{1 + 3 + 6 + ...+ t_k},...\right\},$$ where $t_k$ denotes the $k$th triangular number. The sum of the first $2024$ terms of $s$, denoted $S$, can be expressed as $$S = 3 \left(\frac{1}{2}+\frac{1}{a}-\frac{1}{b}\right),$$ where $a$ and $b$ are positive integers. Find the minimal possible value of $a + b$. [b]p12.[/b] Omar writes the numbers from $1$ to $1296$ on a whiteboard and then converts each of them into base $6$. Find the sum of all of the digits written on the whiteboard (in base $10$), including both the base $10$ and base $6$ numbers. [b]p13.[/b] A mountain number is a number in a list that is greater than the number to its left and right. Let $N$ be the amount of lists created from the integers $1$ - $100$ such that each list only has one mountain number. $N$ can be expressed as $$N = 2^a(2^b - c^2),$$ where $a$, $b$ and $c$ are positive integers and $c$ is not divisible by $2$. Find $a + b+c$. (The numbers at the beginning or end of a list are not considered mountain numbers.)[hide]Original problem was voided because the original format of the answer didn't match the result's format. So I changed it in the wording, in order the problem to be correct[/hide] [b]p14.[/b] A circle $\omega$ with center $O$ has a radius of $25$. Chords $\overline{AB}$ and $\overline{CD}$ are drawn in $\omega$ , intersecting at $X$ such that $\angle BXC = 60^o$ and $AX > BX$. Given that the shortest distance of $O$ with $\overline{AB}$ and $\overline{CD}$ is $7$ and $15$ respectively, the length of $BX$ can be expressed as $x - \frac{y}{\sqrt{z}}$ , where $x$, $y$, and $z$ are positive integers such that $z$ is not divisible by any perfect square. Find $x + y + z.$ [hide]two answers were considered correct according to configuration[/hide] [b]p15.[/b] How many ways are there to split the first $10$ natural numbers into $n$ sets (with $n \ge 1$) such that all the numbers are used and each set has the same average? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].