Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$. [i]Marian Andronache[/i]

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

Positive real numbers $a_1, \dots, a_k, b_1, \dots, b_k$ are given. Let $A = \sum_{i = 1}^k a_i, B = \sum_{i = 1}^k b_i$. Prove the inequality \[ \left( \sum_{i = 1}^k \frac{a_i b_i}{a_i B + b_i A} - 1 \right)^2 \ge \sum_{i = 1}^k \frac{a_i^2}{a_i B + b_i A} \cdot \sum_{i = 1}^k \frac{b_i^2}{a_i B + b_i A}. \]

Find all integer pairs $(a,b)$ such that $a\cdot b$ divides $a^2+b^2+3$.

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$: (a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$, (b) $f(2n) \ge f(n)$ where $Z_{\ge 0}$ is the set of nonnegative integers. Solve the equation $f(n) = 1000$

There are two identical clocks on the wall, one showing the current Moscow time and the other showing current local time. The minimum distance between the ends of their hour hands equals $m$ and the maximum distance equals $M$. Find the distance between the centres of the clocks. (S Fomin, Leningrad)

How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?

Prove that any positive real numbers a,b,c satisfy the inequlaity $$\frac{1}{(a+b)b}+\frac{1}{(b+c)c}+\frac{1}{(c+a)a}\ge \frac{9}{2(ab+bc+ca)}$$ I.Voronovich

Consider positive real numbers $x,y,z$. Prove the inequality $$\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).$$ [i] (Vlad Robu \& Sergiu Novac)[/i]

The function $f$ satisfies the functional equation \[f(x) +f(y) = f(x + y ) - xy - 1\] for every pair $x,~ y$ of real numbers. If $f( 1) = 1$, then the number of integers $n \neq 1$ for which $f ( n ) = n$ is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinite}$

[b]p1.[/b] A man is walking due east at $2$ m.p.h. and to him the wind appears to be blowing from the north. On doubling his speed to $4$ m.p.h. and still walking due east, the wind appears to be blowing from the nortl^eas^. What is the speed of the wind (assumed to have a constant velocity)? [b]p2.[/b] Prove that any triangle can be cut into three pieces which can be rearranged to form a rectangle of the same area. [b]p3.[/b] An increasing sequence of integers starting with $1$ has the property that if $n$ is any member of the sequence, then so also are $3n$ and $n + 7$. Also, all the members of the sequence are solely generated from the first nummber $1$; thus the sequence starts with $1,3,8,9,10, ...$ and $2,4,5,6,7...$ are not members of this sequence. Determine all the other positive integers which are not members of the sequence. [b]p4.[/b] Three prime numbers, each greater than $3$, are in arithmetic progression. Show that their common difference is a multiple of $6$. [b]p5.[/b] Prove that if $S$ is a set of at least $7$ distinct points, no four in a plane, the volumes of all the tetrahedra with vertices in $S$ are not all equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the product of the minimum and maximum values of $\frac{3x+1}{9x^2+6x+2}$.

Given an integer $n \ge 2$, solve in real numbers the system of equations \begin{align*} \max\{1, x_1\} &= x_2 \\ \max\{2, x_2\} &= 2x_3 \\ &\cdots \\ \max\{n, x_n\} &= nx_1. \\ \end{align*}

Determine all real numbers $a$ such that the equation: $$x^8+ax^4+1=0$$ have four real roots that form an arithmetic progression.

Let $n>1$ be a given integer. The Mint issues coins of $n$ different values $a_1, a_2, ..., a_n$, where each $a_i$ is a positive integer (the number of coins of each value is unlimited). A set of values $\{a_1, a_2,..., a_n\}$ is called [i]lucky[/i], if the sum $a_1+ a_2+...+ a_n$ can be collected in a unique way (namely, by taking one coin of each value). (a) Prove that there exists a lucky set of values $\{a_1, a_2, ..., a_n\}$ with $$a_1+ a_2+...+ a_n < n \cdot 2^n.$$ (b) Prove that every lucky set of values $\{a_1, a_2,..., a_n\}$ satisfies $$a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.$$ Proposed by Ilya Bogdanov

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any reals $x \neq y$ the following equality is true: $$f(x+y)^2=f(x+y)+f(x)+f(y)$$ [i]D. Zmiaikou[/i]

Let $a_0$ be an arbitrary positive integer. Consider the infinite sequence $(a_n)_{n\geq 1}$, defined inductively as follows: given $a_0, a_1, ..., a_{n-1}$ define the term $a_n$ as the smallest positive integer such that $a_0+a_1+...+a_n$ is divisible by $n$. Prove that there exist a positive integer a positive integer $M$ such that $a_{n+1}=a_n$ for all $n\geq M$.

Given are $n$ points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with leading coefficient equal to $1$. A parabola is called $good$ if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it). What is the greatest number of $good$ parabolas?

Suppose $a_1,a_2,a_3,a_4$ are distinct integers and $P(x)$ is a polynomial with integer coefficients satisfying $P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1$. (a) Prove that there is no integer $n$ such that $P(n) = 12$. (b) Do there exist such a polynomial and $a_n$ integer $n$ such that $P(n) = 1998$?

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^* $

Determine all real numbers $x$ which satisfy \[ x = \sqrt{a - \sqrt{a+x}} \] where $a > 0$ is a parameter.

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.

Let $P(x)$ be a polynomial with real coefficients so that equation $P(m) + P(n) = 0$ has infinitely many pairs of integer solutions $(m,n)$. Prove that graph of $y = P(x)$ has a center of symmetry.

A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.