A cue ball is shot at a $45$ degree angle from the upper right corner of a billiard table with dimensions $4\text{ ft}$ by $5\text{ ft}$, as shown. How many times does the ball bounce before hitting another corner? Assume that when the ball bounces, its path is perfectly reflected. The final impact in the corner does not count as a bounce. [asy] size(120); draw((0,0)--(5,0)--(5,4)--(0,4)--cycle); label("$5$",(0,0)--(5,0),S); label("$4$",(0,0)--(0,4),W); filldraw(circle((0.4,3.6),0.4),black); draw((0,4)--(1.5,2.5),EndArrow); draw((1.5,2.5)--(4,0)--(5,1), dashed); draw(arc((0,4),1.25,315,270)); label(scale(0.8)*"$45^\circ$",(0.2,2.8),SE); [/asy] $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

Let $ABC$ be a scalene triangle. The incircle is tangent to lines $BC$, $AC$, and $AB$ at points $D$, $E$, and $F$, respectively, and the $A$-excircle is tangent to lines $BC$, $AC$, and $AB$ at points $D_1$, $E_1$, and $F_1$, respectively. Suppose that lines $AD$, $BE$, and $CF$ are concurrent at point $G$, and suppose that lines $AD_1$, $BE_1$, and $CF_1$ are concurrent at point $G_1$. Let line $GG_1$ intersect the internal bisector of angle $BAC$ at point $X$. Suppose that $AX=1$, $\cos{\angle BAC}=\sqrt{3}-1$, and $BC=8\sqrt[4]{3}$. Then $AB \cdot AC = \frac{j+k\sqrt{m}}{n}$ for positive integers $j$, $k$, $m$, and $n$ such that $\gcd(j,k,n)=1$ and $m$ is not divisible by the square of any integer greater than $1$. Compute $1000j+100k+10m+n$. [i]Proposed by Luke Robitaille and Brandon Wang[/i]

Let be the sequence $ \left( a_n \right)_{n\ge 0} $ of positive real numbers defined by $$ a_n=1+\frac{a_{n-1}}{n} ,\quad\forall n\ge 1. $$ Calculate $ \lim_{n\to\infty } a_n ^n . $ [i]Florian Dumitrel[/i]

Let $A$ be the set of all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(xy)=xf(y)$ for all $x,y \in \mathbb{R}$. (a) If $f \in A$ then show that $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$ (b) For $g,h \in A$, define a function $g\circ h$ by $(g \circ h)(x)=g(h(x))$ for $x \in \mathbb{R}$. Prove that $g \circ h$ is in $A$ and is equal to $h \circ g$.

Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.

In a country with $5$ distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities $\{A,B,C\}$ such that there are roads between $A$ and $B$, $B$ and $C$, and $C$ and $A$. How many road systems between the five cities are possible?

Given an $n \times n$ board which is divided into $n^2$ squares of size $1 \times 1$, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one $1\times 2$ domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a $2\times 1$ domino. After Aqua colors, it turns out there are exactly $2024$ ways for Ruby to place a domino on the board so that it covers exactly $1$ black square and $1$ white square. Determine the smallest possible value of $n$ so that Aqua and Ruby can do this. [i]Proposed by Muhammad Afifurrahman, Indonesia [/i]

We are given a $4\times 4$ table. a) Place $7$ stars in the cells in such a way that the erasing of any two rows and two columns will leave at least one of the stars. b) Prove that if there are less than $7$ stars, you can always find two columns and two rows such that erasing them, no star remains in the table.

Of all the fractions $\frac{x}{y}$ that satisfy $$\frac{41}{2010}<\frac{x}{y}<\frac{1}{49}$$ find the one with the smallest denominator.

Allen and Alan play a game. A nonconstant polynomial $P(x,y)$ with real coefficients and a positive integer $d$ greater than the degree of $P$ are known to both Allen and Alan. Alan thinks of a polynomial $Q(x,y)$ with real coefficients and degree at most $d$ and keeps it secret. Allen can make queries of the form $(s,t)$, where $s$ and $t$ are real numbers such that $P(s,t)\neq0$. Alan must respond with the value $Q(s,t)$. Allen's goal is to determine whether $P$ divides $Q$. Find (in terms of $P$ and $d$) the smallest positive integer, $g$, such that Allen can always achieve this goal making no more than $g$ queries. [i]Linus Tang[/i]

The numbers $1,\, 2,\, \ldots\, , n^{2}$ are written in the squares of an $n \times n$ board in some order. Initially there is a token on the square labelled with $n^{2}.$ In each step, the token can be moved to any adjacent square (by side). At the beginning, the token is moved to the square labelled with the number $1$ along a path with the minimum number of steps. Then it is moved to the square labelled with $2,$ then to square $3,$ etc, always taking the shortest path, until it returns to the initial square. If the total trip takes $N$ steps, find the smallest and greatest possible values of $N.$

Moor and Samantha are drinking tea at a constant rate. If Moor starts drinking tea at $8:00$ am, he will finish drinking $7$ cups of tea by $12:00$ pm. If Samantha joins Moor at $10:00$ am, they will finish drinking the $7$ cups of tea by $11:15$ am. How many hours would it take Samantha to drink $1$ cup of tea?

Let $S$ be the set of degree $4$ polynomials $f$ with complex number coefficients satisfying $f(1)=f(2)^2=f(3)^3$ $=$ $f(4)^4=f(5)^5=1.$ Find the mean of the fifth powers of the constant terms of all the members of $S.$

Ayse knows the weights of nine balls with different colors are $1,2,\cdots, 9$ grams, but she doesn't know the weight of a specific ball. But Baris knows the weight of each ball. Baris wants to prove his knowledge to Ayse. There is a double pan balance which shows the heavier pan and the difference of the two pans. At least how many weighs are required for proof of Ali's knowledge? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y,z \in \mathbb{Q}$: \[f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).\]

Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

Find all quadruplets $(a, b, c, d)$ of positive integers such that \[ \left( 1 + \frac{1}{a} \right) \left( 1 + \frac{1}{b} \right) \left( 1 + \frac{1}{c} \right) \left( 1 + \frac{1}{d} \right) = 4. \]

Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

Let $p$ be an odd prime number. Prove that there exists a unique integer $k$ such that $0 \le k \le p^2$ and $p^2$ divides $k(k + 1)(k + 2) ... (k + p - 3) - 1$. Malik Talbi

Two circles $\Gamma_a$ and $\Gamma_b$ with their centres lying on the legs $BC$ and $CA$ of a right triangle, both touching the hypotenuse $AB$, and both passing through the vertex $C$ are given. Let the radii of these circles be denoted by $\gamma_a$ and $\gamma_b$. Find the greatest real number $p$ such that the inequality $\frac{1}{\gamma_a}+\frac{1}{\gamma_b}\ge p \left(\frac{1}{a}+\frac{1}{b}\right)$ ($BC = a,CA = b$) holds for all right triangles $ABC$.

Integers $a_1, a_2, \dots a_n$ are different at $\text{mod n}$. If $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$ are also different at $\text{mod n}$, we call the ordered $n$-tuple $(a_1, a_2, \dots a_n)$ [i]lucky[/i]. For which positive integers $n$, one can find a lucky $n$-tuple?

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

If $ \log_{10}{m} \equal{} b \minus{} \log_{10}{n}$, then $ m$= $\textbf{(A)}\ \dfrac{b}{n} \qquad \textbf{(B)}\ bn \qquad \textbf{(C)}\ 10^b n\qquad \textbf{(D)}\ b-10^n \qquad \textbf{(E)}\ \dfrac{10^b}{n}$

Let $a,b,c$ be positive integers such that $a^2+b^2+c^2$ is divisble by $a+b+c$. Prove that at least two of the numbers $a^3,b^3,c^3$ leave the same remainder by division through $a+b+c$.

[b]p1.[/b] Let $A$ be the point $(-1, 0)$, $B$ be the point $(0, 1)$ and $C$ be the point $(1, 0)$ on the $xy$-plane. Assume that $P(x, y)$ is a point on the $xy$-plane that satisfies the following condition $$d_1 \cdot d_2 = (d_3)^2,$$ where $d_1$ is the distance from $P$ to the line $AB$, $d_2$ is the distance from $P$ to the line $BC$, and $d_3$ is the distance from $P$ to the line $AC$. Find the equation(s) that must be satisfied by the point $P(x, y)$. [b]p2.[/b] On Day $1$, Peter sends an email to a female friend and a male friend with the following instructions: $\bullet$ If you’re a male, send this email to $2$ female friends tomorrow, including the instructions. $\bullet$ If you’re a female, send this email to $1$ male friend tomorrow, including the instructions. Assuming that everyone checks their email daily and follows the instructions, how many emails will be sent from Day $1$ to Day $365$ (inclusive)? [b]p3.[/b] For every rational number $\frac{a}{b}$ in the interval $(0, 1]$, consider the interval of length $\frac{1}{2b^2}$ with $\frac{a}{b}$ as the center, that is, the interval $\left( \frac{a}{b}- \frac{1}{2b^2}, \frac{a}{b}+\frac{1}{2b^2}\right)$ . Show that $\frac{\sqrt2}{2}$ is not contained in any of these intervals. [b]p4.[/b] Let $a$ and $b$ be real numbers such that $0 < b < a < 1$ with the property that $$\log_a x + \log_b x = 4 \log_{ab} x - \left(\log_b (ab^{-1} - 1)\right)\left(\log_a (ab^{-1} - 1) + 2 log_a ab^{-1} \right)$$ for some positive real number $x \ne 1$. Find the value of $\frac{a}{b}$. [b]p5.[/b] Find the largest positive constant $\lambda$ such that $$\lambda a^2 b^2 (a - b)^2 \le (a^2 - ab + b^2)^3$$ is true for all real numbers $a$ and $b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].