Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Ada has been told To write down five haikus plus Two more every hour. Such that she needs to Write down five in the first hour Seven, nine, so on. Ada has so far Forty haikus and writes down Seven every hour. At which hour after She begins will she not have Enough haikus done? [i]Proposed by Ada Tsui[/i]

Some positive integers, sum of which is $23$, are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$. [b]a) [/b]Is it possible that the sequence contains exactly $11$ terms? [b]b)[/b]Is it possible that the sequence contains exactly $12$ terms?

Let $P(x)$ be a polynomial of odd degree. Prove that the equation $P(P(x)) = 0$ has at least as many different real roots as the equation $P(x) = 0$ [hide=original wording]Пусть P(x) — многочлен нечетной степени. Докажите, что уравнение P(P(x)) = 0 имеет не меньше различных действительных корней, чем уравнение P(x) = 0[/hide]

Find all functions $f$ from the set of reals to itself so that for all reals $x,y,$ $$f(x)f(f(x)+y) = f(x^2) + f(xy).$$ [i]Proposed by Culver Kwan[/i]

Isaac writes each fraction $\frac{1^2}{300}$ , $\frac{2^2}{300}$ , $...$, $\frac{300^2}{300}$ in reduced form. Compute the sum of all denominators over all the reduced fractions that Isaac writes down.

Call a polynomial $p(x)$ with positive integer roots [i]corrupt[/i] if there exists an integer that cannot be expressed as a sum of (not necessarily positive) multiples of its roots. The polynomial $A(x)$ is monic, corrupt, and has distinct roots. As well, $A(0)$ has $7$ positive divisors. Find the least possible value of $|A(1)|$.

Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously: (1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$; (2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$; (3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$. Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.

Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule: (a) $g$ is nondecrasing (b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$, Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$

Prove that there exists a function $f : \mathbb{N} \rightarrow \mathbb{N}$ that satisfies the following (1) $\{f(n) : n\in\mathbb{N}\}$ is a finite set; and (2) For nonzero integers $x_1, x_2, \ldots, x_{1000}$ that satisfy $f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)$, then $x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0$.

[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$. [b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$. [b]p3.[/b] For a positive integer $n$, there is a sequence $(a_0, a_1, a_2,..., a_n)$ of real values such that $a_0 = 11$ and $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$ for every $k$ with $0 \le k \le n-1$. Find the maximum possible value of $n$. (Be careful that your answer isn’t off by one!) [b]p4.[/b] Persons $A$ and $B$ stand at point $P$ on line $\ell$. Point $Q$ lies at a distance of $10$ from point $P$ in the direction perpendicular to $\ell$. Both persons intially face towards $Q$. Person $A$ walks forward and to the left at an angle of $25^o$ with $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the right, and continues walking. Person $B$ walks forward and to the right at an angle of $55^o$ with line $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the left, and continues walking. Their paths cross at point $R$. Find the distance $PR$. [b]p5.[/b] Compute $$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$ [b]p6.[/b] There is a unique real value $A$ such that for all $x$ with $1 < x < 3$ and $x \ne 2$, $$\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.$$ Compute $A$. [b]p7.[/b] Nine poles of height $1, 2,..., 9$ are placed in a line in random order. A pole is called [i]dominant [/i] if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly $2$ dominant poles. [b]p8.[/b] $\tan (11x) = \tan (34^o)$ and $\tan (19x) = \tan (21^o)$. Compute $\tan (5x)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Evaluate $$\sum^{\infty}_{k=0} \left( \frac{-1}{8}\right)^k {2k \choose k}$$

The lateral surface of a cylinder of revolution is divided by $n-1$ planes parallel to the base and $m$ parallel generators into $mn$ cases $( n\ge 1,m\ge 3)$. Two cases will be called neighbouring cases if they have a common side. Prove that it is possible to write a real number in each case such that each number is equal to the sum of the numbers of the neighbouring cases and not all the numbers are zero if and only if there exist integers $k,l$ such that $n+1$ does not divide $k$ and \[ \cos \frac{2l\pi}{m}+\cos\frac{k\pi}{n+1}=\frac{1}{2}\] [i]Ciprian Manolescu[/i]

The distance from $A$ to $B$ is $d$ kilometres. A plane flying with the constant speed in the constant direction along and over the line $(AB)$ is being watched from those points. Observers have reported that the angle to the plane from the point $A$ has changed by $\alpha$ degrees and from $B$ --- by $\beta$ degrees within one second. What can be the minimal speed of the plane?

Determine the largest number in the infinite sequence \[ 1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots\]

Consider the polynomials $P(x)=x^{6}-x^{5}-x^{3}-x^{2}-x$ and $Q(x)=x^{4}-x^{3}-x^{2}-1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x)=0,$ find $P(z_{1})+P(z_{2})+P(z_{3})+P(z_{4}).$

Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000 .

Determine the integers $a, b, c$ for which $$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

For this exercise, $ \{\} $ denotes the fractional part. [b]a)[/b] Let be a natural number $ n. $ Compare $ \left\{ \sqrt{n+1} -\sqrt{n} \right\} $ with $ \left\{ \sqrt{n} -\sqrt{n-1} \right\} . $ [b]b)[/b] Show that there are two distinct natural numbers $ a,b, $ such that $ \left\{ \sqrt{a} -\sqrt{b} \right\} =\left\{ \sqrt{b} -\sqrt{a} \right\} . $ [i]Traian Preda[/i]

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

A sequence of numbers $a_1,a_2,...,a_{1999}$ is given. In each move it is allowed to choose two of the numbers, say $a_m,a_n$, and replace them by the numbers $$\frac{a_n^2}{a_m^2}-\frac{n}{m}\left(\frac{a_m^2}{a_n}-a_m\right), \frac{a_m^2}{a_n^2}-\frac{m}{n}\left(\frac{a_n^2}{a_m}-a_n\right) $$ respectively. Starting with the sequence $a_i = 1$ for $20 \nmid i$ and $a_i =\frac{1}{5}$ for $20 \mid i$, is it possible to obtain a sequence whose all terms are integers?