We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

Let $ A$ be a closed and bounded set in the plane, and let $ C$ denote the set of points at a unit distance from $ A$. Let $ p \in C$, and assume that the intersection of $ A$ with the unit circle $ K$ centered at $ p$ can be covered by an arc shorter that a semicircle of $ K$. Prove that the intersection of $ C$ with a suitable neighborhood of $ p$ is a simple arc which $ p$ is not an endpoint. [i]M. Bognar[/i]

The polynomial $P$ of degree $n$ satisfies $P(k) = \frac{k}{k +1}$ for $k = 0,1,2,...,n$. Find $P(n+1)$.

The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square. [img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]

Let $a, b$ and $c$ be positive real numbers such that $abc = 1$. Prove the inequality $\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c)$.

Source: 2018 Canadian Open Math Challenge Part B Problem 4 ----- Determine the number of $5$-tuples of integers $(x_1,x_2,x_3,x_4,x_5)$ such that $\text{(a)}$ $x_i\ge i$ for $1\le i \le 5$; $\text{(b)}$ $\sum_{i=1}^5 x_i = 25$.

In decimal representation $$\text {34!=295232799039a041408476186096435b0000000}.$$ Find the numbers $a$ and $b$.

Prove that there does not exist a polynomial $P (x)$ with integer coefficients and a natural number $m$ such that $$x^m + x + 2 = P(P(x))$$ holds for all integers $x$.

Consider $2002$ segments on a plane, such that their lengths are the same. Prove that there exists such a straight line $r$ such that the sum of the lengths of the projections of the $2002$ segments about $r$ is less than $\frac{2}{3}$.

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

The Konigsberg School has assigned grades $1$ through $7$ to pods $A$ through $G$, one grade per pod. The school noticed that each pair of connected pods has been assigned grades differing by $2$ or more grade levels. (For example, grades $1$ and $2$ will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods $C, E,$ and $F$? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 15\qquad \textbf{(E)}\ 16$\\

In a french village the number of inhabitants is a perfect square. If $100$ more people moved in, then the number of people would be $ 1$ bigger than a perfect square. If again $100$ more people moved in, then the number of people would be a perfect square again. How many people lives in the village if their number is the least possible?

(a) Is it possible to form a prime number using all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 exactly once? (b) Consider the following magic square where the sum of each row, column, and diagonal is the same (in this case, 15): \[ \begin{array}{ccc} 6 & 7 & 2 \\ 1 & 5 & 9 \\ 8 & 3 & 4 \\ \end{array} \] Is it possible to create a magic square with the same properties using the numbers 11, 12, 13, 14, 15, 16, 17, 18, and 19?

Consider the following modified algorithm for binary search, which we will call $\textit{weighted binary search}$: \begin{tabular}{l} 01: \textbf{FUNCTION} SEARCH($L$, value) \\ 02:$\qquad$ hi $\leftarrow$ $\operatorname{len}(L) - 1$ \\ 03:$\qquad$ lo $\leftarrow$ 0 \\ 04:$\qquad$ \textbf{WHILE} hi $\geq$ lo \\ 05:$\qquad\qquad$ guess $\leftarrow$ $\lfloor w \cdot\text{lo} + (1-w) \cdot \text{hi}\rfloor$ \\ 06:$\qquad\qquad$ mid $\leftarrow$ $L[\text{guess}]$ \\ 07:$\qquad\qquad$ \textbf{IF} mid $> \text{value}$ \\ 08: $\qquad\qquad\qquad$ hi $\leftarrow$ $\text{guess} - 1$ \\ 09: $\qquad\qquad$ \textbf{ELSE IF} mid $< \text{value}$ \\ 10: $\qquad\qquad\qquad$ lo $\leftarrow$ $\text{guess} + 1$ \\ 11: $\qquad\qquad$ \textbf{ELSE} \\ 12: $\qquad\qquad\qquad$ \textbf{RETURN} guess \\ 13:$\qquad$ \textbf{RETURN} -1 (not found) \end{tabular}\\ Assume $L$ is a list of the integers $\{1,2,\ldots,100\}$, in that order. Further assume that accessing the $k$th index of $L$ costs $k+1$ tokens (e.g. $L[0]$ costs $1$ token). Let $S$ be the set of all $w\in[\tfrac12,1)$ which minimize the average cost when $\texttt{value}$ is an integer selected at random in the range $[1,50]$. Given that $S=\left(x,\tfrac {74}{99}\right]$, determine $x$.

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves? Proposed by [i]Nikola Velov, Macedonia[/i]

Consider real numbers $x,y,z$ such that $x,y,z>0$. Prove that \[ (xy+yz+xz)\left(\frac{1}{x^2+y^2}+\frac{1}{x^2+z^2}+\frac{1}{y^2+z^2}\right) > \frac{5}{2}. \]

Let $k \in Z_{>0}$ be the smallest positive integer with the property that $k\frac{gcd(x,y)gcd(y,z)}{lcm (x,y^2,z)}$ is a positive integer for all values $1 \le x \le y \le z \le 121$. If k' is the number of divisors of $k$, find the number of divisors of $k'$.

Let S be a set of sequences of length 15 formed by using the letters a and b such that every pair of sequences in S differ in at least 3 places. What is the maximum number of sequences in S?

Consider a $20$-sided convex polygon $K$, with vertices $A_1, A_2,...,A_{20}$ in that order. Find the number of ways in which three sides of $K$ can be chosen so that every pair among them has at least two sides of $K$ between them. (For example $(A_1A_2, A_4A_5, A_{11}A_{12})$ is an admissible triple while $(A_1A_2, A_4A_5, A_{19}A_{20})$ is not.

Prove that for every positive integer $ n$, $ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$

Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $ \Gamma$ touching arcs $AC$ and $BD$ both externally and also touching the side $CD$. Find the radius of $ \Gamma$.

Let N be a positive integer greater than 2. We number the vertices of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N + 1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way. In the first step we mark the vertex 1. If ni is the vertex marked in the i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away from vertex ni, counting clockwise if ni is positive and counter-clockwise if ni is negative. This procedure is repeated till we reach a vertex that has already been marked. Let $f(N)$ be the number of non-marked vertices. (a) If $f(N) = 0$, prove that 2N + 1 is a prime number. (b) Compute $f(1997)$.

a) Do there exist four equilateral triangles in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? (Note that in both parts, there is no assumption on the intersection of interior of polygons.) [i]Proposed by Hesam Rajabzadeh[/i]

The cost of five water bottles is \$13, rounded to the nearest dollar, and the cost of six water bottles is \$16, also rounded to the nearest dollar. If all water bottles cost the same integer number of cents, compute the number of possible values for the cost of a water bottle. [i]Proposed by Eugene Chen[/i]

Six people play the following game: They have a cube, initially white. One by one, the players mark an $ X$ on a white face of the cube, and roll it like a die. The winner is the first person to roll an $ X$ (for example, player 1 wins with probability $ \frac {1}{6}$, while if none of players 1-5 win, player 6 will place an $ X$ on the last square and win for sure). What is the probability that the sixth player wins?