The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies $f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$. (i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$. (ii) Determine the smallest possible $a$.

[u]Round 1[/u] [b]p1.[/b] If this mathathon has $7$ rounds of $3$ problems each, how many problems does it have in total? (Not a trick!) [b]p2.[/b] Five people, named $A, B, C, D,$ and $E$, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started? [b]p3.[/b] At Barrios’s absurdly priced fish and chip shop, one fish is worth $\$13$, one chip is worth $\$5$. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips? [u]Round 2[/u] [b]p4.[/b] If there are $15$ points in $4$-dimensional space, what is the maximum number of hyperplanes that these points determine? [b]p5.[/b] Consider all possible values of $\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4}$ for any distinct complex numbers $z_1$, $z_2$, $z_3$, and $z_4$. How many complex numbers cannot be achieved? [b]p6.[/b] For each positive integer $n$, let $S(n)$ denote the number of positive integers $k \le n$ such that $gcd(k, n) = gcd(k + 1, n) = 1$. Find $S(2015)$. [u]Round 3 [/u] [b]p7.[/b] Let $P_1$, $P_2$,$...$, $P_{2015}$ be $2015$ distinct points in the plane. For any $i, j \in \{1, 2, ...., 2015\}$, connect $P_i$ and $P_j$ with a line segment if and only if $gcd(i - j, 2015) = 1$. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let $\omega$ be the unique positive integer such that there exists a clique with $\omega$ elements and such that there does not exist a clique with $\omega + 1$ elements. Find $\omega$. [b]p8.[/b] A Chinese restaurant has many boxes of food. The manager notices that $\bullet$ He can divide the boxes into groups of $M$ where $M$ is $19$, $20$, or $21$. $\bullet$ There are exactly $3$ integers $x$ less than $16$ such that grouping the boxes into groups of $x$ leaves $3$ boxes left over. Find the smallest possible number of boxes of food. [b]p9.[/b] If $f(x) = x|x| + 2$, then compute $\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k))$. [u]Round 4 [/u] [b]p10.[/b] Let $ABC$ be a triangle with $AB = 13$, $BC = 20$, $CA = 21$. Let $ABDE$, $BCFG$, and $CAHI$ be squares built on sides $AB$, $BC$, and $CA$, respectively such that these squares are outside of $ABC$. Find the area of $DEHIFG$. [b]p11.[/b] What is the sum of all of the distinct prime factors of $7783 = 6^5 + 6 + 1$? [b]p12.[/b] Consider polyhedron $ABCDE$, where $ABCD$ is a regular tetrahedron and $BCDE$ is a regular tetrahedron. An ant starts at point $A$. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After $6$ moves, what is the probability the ant is back at point $A$? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For all positive integers $n,$ let $r_n$ be defined as \[r_n=\sum_{i=0}^n(-1)^i\binom{n}{i}\frac{1}{(i+1)!}.\]Prove that $\sum_{r=1}^\infty r_i=0.$

The monic polynomial $f$ has rational coefficients and is irreducible over the rational numbers. If $f(\sqrt5 +\sqrt2)= 0$, compute $f(f(\sqrt5 -\sqrt2))$. (A polynomial is [i]monic [/i] if its leading coefficient is $1$. A polynomial is [i]irreducible [/i] over the rational numbers if it cannot be expressed as a product of two polynomials with rational coefficients of positive degree. For example, $x^2 - 2$ is irreducible, but $x^2 - 1 = (x + 1)(x - 1)$ is not.)

Find the integer numbers $a, b, c$ such that the function $f: R \to R$, $f(x) = ax^2 +bx + c$ satisfies the equalities : $$f(f(1) ))= f (f(2 ) )= f(f (3 ))$$

Determine all polynomials $P$ with real coeffients such that $1 + P(x) = \frac12 (P(x -1) + P(x + 1))$ for all real $x$.

What is the value of the following expression? $$\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}$$ $\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{9950} \qquad \textbf{(C) } \frac{4780}{4779} \qquad \textbf{(D) } \frac{108}{107} \qquad \textbf{(E) } \frac{81}{80} $

a) Prove that there is not an infinte sequence $(x_n)$, $n=1,2,...$ of positive real numbers satisfying the relation $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}$, $\forall n \in N$ (*) b) Do there exist sequences satisfying (*) and containing arbitrary many terms? I.Voronovich

okay this one is from Prof. Mircea Lascu from Zalau, Romaniaand Prof. V. Cartoaje from Ploiesti, Romania. It goes like this: given being a triangle ABC for every point M inside we construct the points A[size=67]M[/size], B[size=67]M[/size], C[size=67]M[/size] on the circumcircle of the triangle ABC such that A, A[size=67]M[/size], M are collinear and so on. Find the locus of these points M for which the area of the triangle A[size=67]M[/size] B[size=67]M[/size] C[size=67]M[/size] is bigger than the area of the triangle ABC.

Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

A function $g$ is [i]ever more[/i] than a function $h$ if, for all real numbers $x$, we have $g(x) \ge h(x)$. Consider all quadratic functions $f(x)$ such that $f(1) = 16$ and $f(x)$ is ever more than both $(x + 3)^2$ and $x^2 + 9$. Across all such quadratic functions $f$, compute the minimum value of $f(0)$.

Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality: \[ x\plus{}af(y)\leq y\plus{}f(f(x)) \] for all $ x,y\in\mathbb{R}$

Let ${a}$ be the fractional part of the number $a$, that is, $\{a\} = a - [a]$, where$ [a]$ is the integer part of $ a$. (For example, $\{1.7\} = 1.7 -1 = 0.7$,$\{-\sqrt2 \}= -\sqrt2 -(-3) = 3-\sqrt2$.) a) How many solutions does the equation have $$ \{5\{4\{3\{2\{x\}\}\}\}\}=1\,\, ?$$ b) Find its greatest solution.

We define a sequence with $a_1=850$ and $$a_{n+1}=\frac{a_n^2}{a_n-1}$$ for $n\geq 1$. Find all values of $n$ for which $\lfloor a_n\rfloor =2024$.

Let $a,b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that \[\dfrac{1}{a^3+2b^2+2b+4}+\dfrac{1}{b^3+2c^2+2c+4}+\dfrac{1}{c^3+2a^2+2a+4}\leq \dfrac13.\]

Prove that for all positive integers $n$ holds that $$\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+...+\frac{1}{(2n-1) \cdot 2n}=\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}$$

Show that if $a \ge b \ge c \ge 0$ then $$a^2b(a - b) + b^2c(b - c) + c^2a(c - a) \ge 0.$$

[b]p1.[/b] What is $7\%$ of one half of $11\%$ of $20000$ ? [b]p2.[/b] Three circles centered at $A, B$, and $C$ are tangent to each other. Given that $AB = 8$, $AC = 10$, and $BC = 12$, find the radius of circle $ A$. [b]p3. [/b]How many positive integer values of $x$ less than $2012$ are there such that there exists an integer $y$ for which $\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}$ ? [b]p4. [/b]The positive difference between $ 8$ and twice $x$ is equal to $11$ more than $x$. What are all possible values of $x$? [b]p5.[/b] A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? [b]p6.[/b] A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? [b]p7.[/b] $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH +LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? [b]p8. [/b]A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square? [b]p9.[/b] Consider the sequence $2012, 12012, 20120, 20121, ...$. This sequence is the increasing sequence of all integers that contain “$2012$”. What is the $30$th term in this sequence? [b]p10.[/b] What is the coefficient of the $x^5$ term in the simplified expansion of $(x +\sqrt{x} +\sqrt[3]{x})^{10}$ ? PS. You had better use hide for answers.

Let $(a_n)_{n\ge 1}$ be a sequence of positive numbers. If there is a constant $M > 0$ such that $a_2^2 + a_2^2 +\ldots + a_n^2 < Ma_{n+1}^2$ for all $n$, then prove that there is a constant $M ' > 0$ such that $a_1 + a_2 +\ldots + a_n < M ' a_{n+1}$ .

[i]Round 5[/i] [b]p13.[/b] Kelvin Amphibian, a not-frog who lives on the coordinate plane, likes jumping around. Each step, he jumps either to the spot that is $1$ unit to the right and 2 units up, or the spot that is $2$ units to the right and $1$ unit up, from his current location. He chooses randomly among these two choices with equal probability. He starts at the origin and jumps for a long time. What is the probability that he lands on $(10, 8)$ at some time in his journey? [b]p14.[/b] Points $A, B, C$, and $D$ are randomly chosen on the circumference of a unit circle. What is the probability that line segments $AB$ and $CD$ intersect inside the circle? [b]p15.[/b] Let $P(x)$ be a quadratic polynomial with two consecutive integer roots. If it is also known that $\frac{P(2017)} {P(2016)} = \frac{2016}{2017}$ , find the larger root of $P(x)$. [u]Round 6[/u] [b]p16.[/b] Let $S_n$ be the sum of reciprocals of the integers between $1$ and $n$ inclusive. Find a triple $(a, b, c)$ of positive integers such that $S_{2017} \cdot S_{2017} - S_{2016} \cdot S_{2018} = \frac{S_a+S_b}{c}$ . [b]p17.[/b] Suppose that $m$ and $n$ are both positive integers. Alec has $m$ standard $6$-sided dice, each labelled $1$ to $6$ inclusive on the sides, while James has $n$ standard $12$-sided dice, each labelled $1$ to $12$ inclusive on the sides. They decide to play a game with their dice. They each toss all their dice simultaneously and then compute the sum of the numbers that come up on their dice. Whoever has a higher sum wins (if the sums are equal, they tie). Given that both players have an equal chance of winning, determine the minimum possible value of mn. [b]p18.[/b] Overlapping rectangles $ABCD$ and $BEDF$ are congruent to each other and both have area $1$. Given that $A,C,E, F$ are the vertices of a square, find the area of the square. [u]Round 7[/u] [b]p19.[/b] Find the number of solutions to the equation $$||| ... |||||x| + 1| - 2| + 3| - 4| +... - 98| + 99| - 100| = 0$$ [b]p20.[/b] A split of a positive integer in base $10$ is the separation of the integer into two nonnegative integers, allowing leading zeroes. For example, $2017$ can be split into $2$ and $017$ (or $17$), $20$ and $17$, or $201$ and $7$. A split is called squarish if both integers are nonzero perfect squares. $49$ and $169$ are the two smallest perfect squares that have a squarish split ($4$ and $9$, $16$ and $9$ respectively). Determine all other perfect squares less than $2017$ with at least one squarish split. [b]p21.[/b] Polynomial $f(x) = 2x^3 + 7x^2 - 3x + 5$ has zeroes $a, b$ and $c$. Cubic polynomial $g(x)$ with $x^3$-coefficient $1$ has zeroes $a^2$, $b^2$ and $c2$. Find the sum of coefficients of $g(x)$. [u]Round 8[/u] [b]p22.[/b] Two congruent circles, $\omega_1$ and $\omega_2$, intersect at points $A$ and $B$. The centers of $\omega_1$ and $\omega_2$ are $O_1$ and $O_2$ respectively. The arc $AB$ of $\omega_1$ that lies inside $\omega_2$ is trisected by points $P$ and $Q$, with the points lying in the order $A, P, Q,B$. Similarly, the arc $AB$ of $\omega_2$ that lies inside $\omega_1$ is trisected by points $R$ and $S$, with the points lying in the order $A,R, S,B$. Given that $PQ = 1$ and $PR =\sqrt2$, find the measure of $\angle AO_1B$ in degrees. [b]p23.[/b] How many ordered triples of $(a, b, c)$ of integers between $-10$ and $10$ inclusive satisfy the equation $-abc = (a + b)(b + c)(c + a)$? [b]p24.[/b] For positive integers $n$ and $b$ where $b > 1$, define $s_b(n)$ as the sum of digits in the base-$b$ representation of $n$. A positive integer $p$ is said to dominate another positive integer $q$ if for all positive integers $n$, $s_p(n)$ is greater than or equal to $s_q(n)$. Find the number of ordered pairs $(p, q)$ of distinct positive integers between $2$ and $100$ inclusive such that $p$ dominates $q$. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2936487p26278546]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

Find the smallest and largest values of the expression $$\frac{ \left| ...\left| |x-1|-1\right| ... -1\right| +1}{\left| |x-2|-1 \right|+1}$$ (The number of units in the numerator of a fraction, including the last one, is eleven, of which ten are under the absolute value sign.)

Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: [i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order; [i](ii)[/i] the polygon is circumscribable about a circle. [i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.

Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

It is known that $x^2_1+ x^2_2+...+ x^2_6= 6$ and $x_1 + x_2 +....+ x_6 = 0.$ Prove that $ x_1x_2....x_6 \le \frac12$ . .