Solve in the set of real numbers the equation $$ 16\{ x \}^2-8x=-1, $$ where $ \{\} $ denotes the fractional part.

The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$.

Let $a, b, c$ be positive real numbers such that $$a^2 + b^2 + c^2 = \frac{1}{4}.$$ Prove that $$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$ [i]Proposed by Petar Filipovski, Macedonia[/i]

Let $a,b,c>0$ such that $a+b+c=3$. Prove that :$$\frac{ab}{ab+a+b}+\frac{bc}{bc+b+c}+\frac{ca}{ca+c+a}+\frac{1}{9}\left(\frac{(a-b)^2}{ab+a+b}+\frac{(b-c)^2}{bc+b+c}+\frac{(c-a)^2}{ca+c+a}\right)\leq1.$$

Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively. Suppose that $\frac{ax + b}{cx + d} \in Q$ for every $x \in N^*$: Prove that there exist integers $A,B,C,D$ such that $\frac{ax + b}{cx + d}= \frac{Ax + B}{Cx+D}$ for all $x \in N^* $

Let $a,b,$ and $c$ be pairwise distinct complex numbers such that $$a^2=b+6, \quad b^2=c+6, \quad \text{and} \quad c^2=a+6.$$ Compute the two possible values of $a+b+c.$

[b]Problem Section #2 c). Denote by $\mathbb{Q^+}$ the set of all positive rational numbers. Determine all functions $f:\mathbb{Q^+}\to\mathbb{Q^+}$ which satisfy the following equation for all $x,y \in \mathbb{Q^+} : f(f(x)^2.y)=x^3.f(xy)$.

For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by \[ a\ast b=\frac{a+b-2ab}{1-ab}.\] Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. $ a.$ Prove that this single number is the same regardless of the choice of pair at each stage. $ b.$ Suppose that the condition on the numbers $ x$ is weakened to $ 0<x\leq 1$. What happens if the list contains exactly one $ 1$?

Prove that for any natural number $n$ the following inequality holds $$4^n < (2n+1)C_{2n}^n$$

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

Find the positive integer $n$ such that $32$ is the product of the real number solutions of $x^{\log_2(x^3)-n} = 13$

Solve over non-negative integers the system $$ \begin{cases} x+y+z^2=xyz, \\ z\leq min(x,y). \end{cases} $$

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that: - $f$ is strictly increasing, - there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$, - for every $x \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$ such that \[ f(y) = \frac{f(x) + f(x + 2024)}{2}. \] [i]Proposed by Pavle Martinović[/i]

Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$ Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.

If $x,y,z,p,q,r$ are real numbers such that \[\frac{1}{x+p}+\frac{1}{y+p}+\frac{1}{z+p}=\frac{1}{p}\]\[\frac{1}{x+q}+\frac{1}{y+q}+\frac{1}{z+q}=\frac{1}{q}\]\[\frac{1}{x+r}+\frac{1}{y+r}+\frac{1}{z+r}=\frac{1}{r}.\]Find the numerical value of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$.

Prove that if the real numbers $ a $, $ b $, $ c $ satisfy the inequalities $$a + b + c> 0,$$ $$ ab + bc + ca > 0$$ $$ abc > 0$$ then $a > 0, b > 0, c > 0$.

Which of the following rationals is greater , $\frac{1995^{1994} + 1}{1995^{1995} + 1}$ or $\frac{1995^{1995} + 1}{ 1995^{1996} +1}$ ?

A function $f : [0,1] \to R$ has the following properties: (a) $f(x) \ge 0$ for $0 < x < 1$, (b) $f(1) = 1$, (c) $f(x+y) \ge f(x)+ f(y) $ whenever $x,y,x+y \in [0,1]$. Prove that $f(x) \le 2x$ for all $x \in [0,1]$.

Let $a,b,c,d$ real numbers such that $a+b+c+d=19$ and $a^2+b^2+c^2+d^2=91$. Find the maximum value of \[ \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} \]

A polynomial $P$ has integral coefficients, and it has at least 9 different integral roots. Let $n$ be an integer such that $|P(n)| < 2017$. Prove that $P(n) = 0$.

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and \[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\] for every natural number $m > 0$ and for every integer $n$. Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$

* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.

[u]Round 5[/u] [b]p13.[/b] In coordinate space, a lattice point is a point all of whose coordinates are integers. The lattice points $(x, y, z)$ in three-dimensional space satisfying $0 \le x, y, z \le 5$ are colored in n colors such that any two points that are $\sqrt3$ units apart have different colors. Determine the minimum possible value of $n$. [b]p14.[/b] Determine the number of ways to express $121$ as a sum of strictly increasing positive Fibonacci numbers. [b]p15.[/b] Let $ABCD$ be a rectangle with $AB = 7$ and $BC = 15$. Equilateral triangles $ABP$, $BCQ$, $CDR$, and $DAS$ are constructed outside the rectangle. Compute the area of quadrilateral $P QRS$. [u] Round 6[/u] Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [b]p16.[/b] Let $C$ be the answer to problem $18$. Suppose that $x$ and $y$ are real numbers with $y > 0$ and $$x + y = C$$ $$x +\frac{1}{y} = -2.$$ Compute $y +\frac{1}{y}$. [b]p17.[/b] Let $A$ be the answer to problem $16$. Let $P QR$ be a triangle with $\angle P QR = 90^o$, and let $X$ be the foot of the perpendicular from point $Q$ to segment $P R$. Given that $QX = A$, determine the minimum possible area of triangle $PQR$. [b]p18.[/b] Let $B$ be the answer to problem $17$ and let $K = 36B$. Alice, Betty, and Charlize are identical triplets, only distinguishable by their hats. Every day, two of them decide to exchange hats. Given that they each have their own hat today, compute the probability that Alice will have her own hat in $K$ days. [u]Round 7[/u] [b]p19.[/b] Find the number of positive integers a such that all roots of $x^2 + ax + 100$ are real and the sum of their squares is at most $2013$. [b]p20.[/b] Determine all values of $k$ such that the system of equations $$y = x^2 - kx + 1$$ $$x = y^2 - ky + 1$$ has a real solution. [b]p21.[/b] Determine the minimum number of cuts needed to divide an $11 \times 5 \times 3$ block of chocolate into $1\times 1\times 1$ pieces. (When a block is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.) [u]Round 8[/u] [b]p22.[/b] A sequence that contains the numbers $1, 2, 3, ... , n$ exactly once each is said to be a permutation of length $n$. A permutation $w_1w_2w_3... w_n$ is said to be sad if there are indices $i < j < k$ such that $w_j > w_k$ and $w_j > w_i$. For example, the permutation $3142756$ is sad because $7 > 6$ and $7 > 1$. Compute the number of permutations of length $11$ that are not sad. [b]p23.[/b] Let $ABC$ be a triangle with $AB = 39$, $BC = 56$, and $CA = 35$. Compute $\angle CAB - \angle ABC$ in degrees. [b]p24.[/b] On a strange planet, there are $n$ cities. Between any pair of cities, there can either be a one-way road, two one-way roads in different directions, or no road at all. Every city has a name, and at the source of every one-way road, there is a signpost with the name of the destination city. In addition, the one-way roads only intersect at cities, but there can be bridges to prevent intersections at non-cities. Fresh Mann has been abducted by one of the aliens, but Sophy Moore knows that he is in Rome, a city that has no roads leading out of it. Also, there is a direct one-way road leading from each other city to Rome. However, Rome is the secret police’s name for the so-described city; its official name, the name appearing on the labels of the one-way roads, is unknown to Sophy Moore. Sophy Moore is currently in Athens and she wants to head to Rome in order to rescue Fresh Mann, but she does not know the value of $n$. Assuming that she tries to minimize the number of roads on which she needs to travel, determine the maximum possible number of roads that she could be forced to travel in order to find Rome. Express your answer as a function of $n$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2809419p24782489]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What can be the angle between the hour and minute hands of a clock if it is known that its value has not changed after $30$ minutes?