Let $C$ be a fixed circle, $u > 0$ be a fixed real and let $v_0 , v_1 , v_2 , \ldots$ be a sequence of positive real numbers. Two ants $A$ and $B$ walk around the perimeter of $C$ in opposite directions, starting from the same starting point. Ant $A$ has a constant speed $u$, while ant $B$ has an initial speed $v_0$. For each positive integer $n$, when the two ants collide for the $n$−th time, they change the directions in which they walk around the perimeter of $C$, with ant $A$ remaining at speed $u$ and ant $B$ stops walking at speed $v_{n-1}$ to walk at speed $v_n$. (a) If the sequence $\{v_n\}$ is strictly increasing, with $\lim_{n\rightarrow \infty} v_n = +\infty$, prove that there is exactly one point in $C$ that ant $A$ will pass "infinitely" many times. (b) Prove that there is a sequence $\{v_n\}$ with $\lim_{n\rightarrow\infty} v_n = +\infty$, such that ant $A$ will pass "infinitely" many times through all points on the circle $C$.

Show that if $(a_n)$ is an infinite sequence of distinct positive integers, neither of which contains digit $0$ in the decimal expansion, then $$\sum_{n=1}^{\infty} \frac{1}{a_n}< 29.$$

The set $C$ of all complex numbers $z$ satisfying $(z +1)^2 = az$ for some $a \in [-10,3]$ is the union of two curves intersecting at a single point in the complex plane. If the sum of the lengths of these two curves is $\ell,$ find $\lfloor \ell \rfloor.$

Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals. Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded? [i]Proposed by Mihai Bălună, Romania[/i]

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Matt has a twenty dollar bill and buys two items worth $\$7:99$ each. How much change does he receive, in dollars? [b]p2.[/b] The sum of two distinct numbers is equal to the positive difference of the two numbers. What is the product of the two numbers? [b]p3.[/b] Evaluate $$\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}.$$ [b]p4.[/b] A sphere with radius $r$ has volume $2\pi$. Find the volume of a sphere with diameter $r$. [b]p5.[/b] Yannick ran $100$ meters in $14.22$ seconds. Compute his average speed in meters per second, rounded to the nearest integer. [b]p6.[/b] The mean of the numbers $2, 0, 1, 5,$ and $x$ is an integer. Find the smallest possible positive integer value for $x$. [b]p7.[/b] Let $f(x) =\sqrt{2^2 - x^2}$. Find the value of $f(f(f(f(f(-1)))))$. [b]p8.[/b] Find the smallest positive integer $n$ such that $20$ divides $15n$ and $15$ divides $20n$. [b]p9.[/b] A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ . [b]p10.[/b] Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl? [b]p11.[/b] Let $k$ be a positive integer. After making $k$ consecutive shots successfully, Andy's overall shooting accuracy increased from $65\%$ to $70\%$. Determine the minimum possible value of $k$. [b]p12.[/b] In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $ \angle AMN = \angle MNP = 90^o$. Compute the ratio $\frac{AP}{PB}$ . [b]p13.[/b] Meena writes the numbers $1, 2, 3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written? [b]p14.[/b] Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube. [b]p15.[/b] A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons? [b]p16.[/b] All positive integers relatively prime to $2015$ are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}-1$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$. [b]p17.[/b] Five red lines and three blue lines are drawn on a plane. Given that $x$ pairs of lines of the same color intersect and $y$ pairs of lines of different colors intersect, find the maximum possible value of $y - x$. [b]p18.[/b] In triangle $ABC$, where $AC > AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are $5$ and $6$, respectively, compute the area of triangle $ABC$. [b]p19.[/b] For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x, y \le 10$ is $(x + y)^2 + (xy - 1)^2$ a prime number? [b]p20.[/b] A solitaire game is played with $8$ red, $9$ green, and $10$ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $c$, he gains a number of points equal to the number of cards that are not of color $c$ in his hand. Find the maximum number of points that he can obtain by the end of the game. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(x, y)$ and $Q(x, y)$ be polynomials in two variables with integer coefficients. The sequences of integers $a_0, a_1,\ldots$ and $b_0, b_1,\ldots$ satisfy \[a_{n+1}=P(a_n,b_n),\quad b_{n+1}=Q(a_n,b_n)\]for all $n\geqslant 0$. Let $m_n$ be the number of integer points of the coordinate plane, lying strictly inside the segment with endpoints $(a_n,b_n)$ and $(a_{n+1},b_{n+1})$. Prove that the sequence $m_0,m_1,\ldots$ is non-decreasing.

Let the polynomials $ w_n $ be given by the formulas: $$ w_1(x) = x^2 - 1, \quad w_{n+1}(x) = w_n(x)^2 - 1, \quad (n = 1, 2, \ldots)$$ and let $a$ be a real number. How many different real solutions does the equation $ w_n(x) = a $ have?

Let $t\ge\frac{1}{2}$ be a real number and $n$ a positive integer. Prove that \[t^{2n}\ge (t-1)^{2n}+(2t-1)^n\]

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that \[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\] for all $x,y>0.$

[b]p1.[/b] Find the smallest positive multiple of $9$ whose digits are all even. [b]p2.[/b] Anika writes down a positive real number $x$ in decimal form. When Nat erases everything to the left of the decimal point, the remaining value is one-fifth of x. Find the sum of all possible values of $x$. [b]p3.[/b] A star-like shape is formed by joining up the midpoints and vertices of a unit square, as shown in the diagram below. Compute the area of this shape. [img]https://cdn.artofproblemsolving.com/attachments/4/8/923b1bf26f6e9872b596e8c81ad1872137f362.png[/img] [b]p4.[/b] Benny and Daria are running a $200$ meter foot race, each at a different constant speed. When Daria finishes the race, she is $14$ meters ahead of Benny. The next time they race, Daria starts 14 meters behind Benny, who starts at the starting line. Both runners run at the same constant speed as in the first race. When Daria reaches the finish line, compute, in centimeters, how far she is ahead of Benny. [b]p5.[/b] In one semester, Ronald takes ten biology quizzes, earning a distinct integer score from $91$ to $100$ on each quiz. He notices that after the first three quizzes, the average of his three most recent scores always increased. Compute the number of ways Ronald could have earned the ten quiz scores. [b]p6.[/b] Ant and Ben are playing a game with stones. They begin with $Z$ stones on the ground. Ant and Ben take turns removing a prime number of stones. Ant moves first. The player who is first unable to make a valid move loses. Find the sum of all positive integers $Z \le 30$ such that Ben can guarantee a win with perfect play. [b]p7.[/b] Let $P$ be a point in a regular octagon as shown in the diagram below. The areas of three triangles are shown. Find the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/0/9/6fde77eeafd04614046292175e4b1411158e85.png[/img] [b]p8.[/b] Find the number of ordered triples $(a, b, c)$ of nonnegative integers with $a \le b \le c$ for which $5a + 4b + 6c = 1200$. [b]p9.[/b] Define $$f(x) = \begin{cases} 2x \,\,\,\, ,\,\,\,\, 0 \le x < \frac12 \\ 2 - 2x \,\,\,\, , \,\,\,\, \frac12 \le x \le 1 \end{cases}$$ Michael picks a real number $0 \le x \le 1$. Michael applies $f$ repeatedly to $x$ until he reaches $x$ again. Find the number of real numbers $x$ for which Michael applies $f$ exactly $12$ times. [b]p10.[/b] In $\vartriangle ABC$, let point $H$ be the intersection of its altitudes and let $M$ be the midpoint of side $BC$. Given that $BC = 4$, $MA = 3$, and $\angle HMA = 60^o$, find the circumradius of $\vartriangle ABC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that for any positive reals $w, x, y, z$ we have $$\sqrt{\frac{w^2 + x^2 + y^2 + z^2}{4}}\ge \sqrt[3]{ \frac{wxy + wxz + wyz + xyz}{4}}$$

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{R}-\{0\}$, $$ f(x) \neq 0 \text{ and } \frac{f(x)}{f(y)} + \frac{f(y)}{f(x)} - f \left( \frac{x}{y}-\frac{y}{x} \right) =2 $$

Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation \[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\] where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]

For any natural $n$ prove the inequality $$\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3$$

Find all reals $z$ such that $z^4 - z^3 - 2z^2 - 3z - 1= 0$.

Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself). Show that $f(1)=0$.

Show that all complex roots of the polynomial $P(z)=a_0z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n$, where $0<a_0<\ldots<a_n$, satisfy $|z|>1$.

Find all functions $u : R \to R$ for which there is a strictly increasing function $f : R \to R$ such that $f(x+y) = f(x)u(y)+ f(y)$ for all $x,y \in R$.

Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.

For what positive integers $n, k$ (with $k < n$) are the binomial coefficients $${n \choose k- 1} \,\,\, , \,\,\, {n \choose k} \,\,\, , \,\,\, {n \choose k + 1}$$ three successive terms of an arithmetic progression?

If $4^{4^4} =\sqrt[128]{2^{2^{2^n}}}$ , find $n$.

For real numbers $a_0,a_1,\cdots,a_n(a_0\neq a_1)$, we have$a_{i-1}+a_{i+1}=2a_i$ for $i=1,2,\cdots,n-1$. Prove that $P(x)=a_0\text{C}_n^0(1-x)^n+a_1\text{C}_n^1x(1-x)^{n-1}+\cdots+a_n\text{C}_n^nx^n$ is a linear polynomial.

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012$.

[u]Round 9[/u] [b]p25.[/b] Let $a$, $b$, and $c$ be positive numbers with $a +b +c = 4$. If $a,b,c \le 2$ and $$M =\frac{a^3 +5a}{4a^2 +2}+\frac{b^3 +5b}{4b^2 +2}+\frac{c^3 +5c}{4c^2 +2},$$ then find the maximum possible value of $\lfloor 100M \rfloor$. [b]p26.[/b] In $\vartriangle ABC$, $AB = 15$, $AC = 16$, and $BC = 17$. Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $CE = 1$ and $BF = 3$. A point $D$ is chosen on side $BC$, and let the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect at point $P \ne D$. Given that $\angle PEF = 30^o$, the length of segment $PF$ can be expressed as $\frac{m}{n}$ . Find $m+n$. [b]p27.[/b] Arnold and Barnold are playing a game with a pile of sticks with Arnold starting first. Each turn, a player can either remove $7$ sticks or $13$ sticks. If there are fewer than $7$ sticks at the start of a player’s turn, then they lose. Both players play optimally. Find the largest number of sticks under $200$ where Barnold has a winning strategy [u]Round 10[/u] [b]p28.[/b] Let $a$, $b$, and $c$ be positive real numbers such that $\log_2(a)-2 = \log_3(b) =\log_5(c)$ and $a +b = c$. What is $a +b +c$? [b]p29.[/b] Two points, $P(x, y)$ and $Q(-x, y)$ are selected on parabola $y = x^2$ such that $x > 0$ and the triangle formed by points $P$, $Q$, and the origin has equal area and perimeter. Find $y$. [b]p30.[/b] $5$ families are attending a wedding. $2$ families consist of $4$ people, $2$ families consist of $3$ people, and $1$ family consists of $2$ people. A very long row of $25$ chairs is set up for the families to sit in. Given that all members of the same family sit next to each other, let the number of ways all the people can sit in the chairs such that no two members of different families sit next to each other be $n$. Find the number of factors of $n$. [u]Round 11[/u] [b]p31.[/b] Let polynomial $P(x) = x^3 +ax^2 +bx +c$ have (not neccessarily real) roots $r_1$, $r_2$, and $r_3$. If $2ab = a^3 -20 = 6c -21$, then the value of $|r^3_1+r^3_2+r^3_3|$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. [b]p32.[/b] In acute $\vartriangle ABC$, let $H$, $I$ , $O$, and $G$ be the orthocenter, incenter, circumcenter, and centroid of $\vartriangle ABC$, respectively. Suppose that there exists a circle $\omega$ passing through $B$, $I$ , $H$, and $C$, the circumradius of $\vartriangle ABC$ is $312$, and $OG = 80$. Let $H'$, distinct from $H$, be the point on $\omega$ such that $\overline{HH'}$ is a diameter of $\omega$. Given that lines $H'O$ and $BC$ meet at a point $P$, find the length $OP$. [b]p33.[/b] Find the number of ordered quadruples $(x, y, z,w)$ such that $0 \le x, y, z,w \le 1000$ are integers and $$x!+ y! =2^z \cdot w!$$ holds (Note: $0! = 1$). [u]Round 12[/u] [b]p34.[/b] Let $Z$ be the product of all the answers from the teams for this question. Estimate the number of digits of $Z$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- |A-E| \rceil \right).$$ Your answer must be a positive integer. [b]p35.[/b] Let $N$ be number of ordered pairs of positive integers $(x, y)$ such that $3x^2 -y^2 = 2$ and $x < 2^{75}$. Estimate $N$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- 2|A-E| \rceil \right).$$ [b]p36.[/b] $30$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \left \lceil 15- \ln \frac{A}{E} \right \rceil \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions: (1) $ f(mn) \equal{} f(m)\plus{}f(n)$, (2) $ f(2008) \equal{} 0$, and (3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$.