Let $a,b,c$ be nonnegative real numbers.Prove that $3(a^2+b^2+c^2)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (a+b+c)^2$.

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

Determine all real polynomials $P$ of degree at most $22$ for which $$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$ for all $k \in \{1, 2, 3, . . . , 21, 22\}$.

Determine the number of real roots of the equation $$|x|+|x+1|+\cdots+|x+2018|=x^2+2018x-2019$$

Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.

[b]p1.[/b] The case of Mr. Brown, Mr. Potter, and Mr. Smith is investigated. One of them has committed a crime. Everyone of them made two statements. Mr. Brown: I have not done it. Mr. Potter has not done it. Mr. Potter: Mr. Brown has not done it. Mr. Smith has done it. Mr. Smith: I have not done it. Mr. Brown has done it. It is known that one of them told the truth both times, one lied both times, and one told the truth one time and lied one time. Who has committed the crime? [b]p2.[/b] Is it possible to draw in a plane $1000001$ circles of the radius $1$ such that every circle touches exactly three other circles? [b]p3.[/b] Consider a circle of radius $R$ centered at the origin. A particle is “launched” from the $x$-axis at a distance, $d$, from the origin with $0 < d < R$, and at an angle, $\alpha$, with the $x$-axis. The particle is reflected from the boundary of the circle so that the [b]angle of incidence[/b] equals the [b]angle of reflection[/b]. Determine the angle $\alpha$ so that the path of the particle contacts the circle’s interior at exactly eight points. Please note that $\alpha$ should be determined in terms of the qunatities $R$ and $d$. [img]https://cdn.artofproblemsolving.com/attachments/e/3/b8ef9bb8d1b54c263bf2b68d3de60be5b41ad0.png[/img] [b]p4.[/b] Is it possible to find four different real numbers such that the cube of every number equals the square of the sum of the three others? [b]p5. [/b]The Fibonacci sequence $(1, 2, 3, 5, 8, 13, 21, . . .)$ is defined by the following formula: $f_n = f_{n-2} + f_{n-1}$, where $f_1 = 1$, $f_2 = 2$. Prove that any positive integer can be represented as a sum of different members of the Fibonacci sequence. [b]p6.[/b] $10,000$ points are arbitrary chosen inside a square of area $1$ m$^2$ . Does there exist a broken line connecting all these points, the length of which is less than $201$ m$^2? PS. You should use hide for answers.

Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form $ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$, if it is known that all the roots of them are positive reals. [i]Proposer[/i]: [b]Baltag Valeriu[/b]

$$(b+c)x^2+(a+c)x+(a+b)=0$$ has not real roots. Prove that $$4ac-b^2 \leq 3a(a+b+c)$$

Consider all sets $A$ of one hundred different natural numbers with the property that any three elements $a,b,c \in A$ (not necessarily different) are the sides of a non-obtuse triangle. Denote by $S(A)$ the sum of the perimeters of all such triangles. Compute the smallest possible value of $S(A)$.

There are $11$ integers (not necessarily distinct) written on the board. Can it turn out that the product of any five of them is greater than the product of the other six?

Let $n$ be a positive integer. Prove that at least one of the integers $[2^n \cdot \sqrt2]$, $[2^{n+1} \cdot \sqrt2]$, $...$, $[2^{2n} \cdot \sqrt2]$ is even, where $[a]$ denotes the integer part of $a$.

In an infinite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows: [list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$ - and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list] Prove that in this sequence [b](a) [/b] the number $1$ occurs infinitely many times; [b](b)[/b] each positive integer occurs infinitely many times. (The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )

[b]p1.[/b] A rectangle $ABCD$ is cut from a piece of paper and folded along a straight line so that the diagonally opposite vertices $A$ and $C$ coincide. Find the length of the resulting crease in terms of the length ($\ell$) and width ($w$) of the rectangle. (Justify your answer.) [b]p2.[/b] The residents of Andromeda use only bills of denominations $\$3 $and $\$5$ . All payments are made exactly, with no change given. What whole-dollar payments are not possible? (Justify your answer.) [b]p3.[/b] A set consists of $21$ objects with (positive) weights $w_1, w_2, w_3, ..., w_{21}$ . Whenever any subset of $10$ objects is selected, then there is a subset consisting of either $10$ or $11$ of the remaining objects such that the two subsets have equal fotal weights. Find all possible weights for the objects. (Justify your answer.) [b]p4.[/b] Let $P(x) = x^3 + x^2 - 1$ and $Q(x) = x^3 - x - 1$ . Given that $r$ and $s$ are two distinct solutions of $P(x) = 0$ , prove that $rs$ is a solution of $Q(x) = 0$ [b]p5.[/b] Given: $\vartriangle ABC$ with points $A_1$ and $A_2$ on $BC$ , $B_1$ and $B_2$ on $CA$, and $C_1$ and $C_2$ on $AB$. $A_1 , A_2, B_1 , B_2$ are on a circle, $B_1 , B_2, C_1 , C_2$ are on a circle, and $C_1 , C_2, A_1 , A_2$ are on a circle. The centers of these circles lie in the interior of the triangle. Prove: All six points $A_1$ , $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ are on a circle. [img]https://cdn.artofproblemsolving.com/attachments/7/2/2b99ddf4f258232c910c062e4190d8617af6fa.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a certain complex number $c$, the polynomial \[ P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\] has exactly 4 distinct roots. What is $|c|$? $\textbf{(A) } 2 \qquad \textbf{(B) } \sqrt{6} \qquad \textbf{(C) } 2\sqrt{2} \qquad \textbf{(D) } 3 \qquad \textbf{(E) } \sqrt{10}$

$f$ is a function twice differentiable on $[0,1]$ and such that $f''$ is continuous. We suppose that : $f(1)-1=f(0)=f'(1)=f'(0)=0$. Prove that there exists $x_0$ on $[0,1]$ such that $|f''(x_0)| \geq 4$

Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\ b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\ c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\ d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

Find all the positive integers $a,b,c$ such that $3ab= 2c^2$ and $a^3+b^3+c^3$ is the double of a prime number.

Prove that the polynomial $P(x) = x^5-x+a$ is irreducible over $\mathbb{Z}$ if $5 \nmid a$.

Determine a geometric progression of seven terms, knowing the sum, $7$, of the first three, and the sum, $112$, of the last three.

A Mathlon is a competition where there are $M$ athletic events. $A, B$ and $C$ were the only participants of a Mathlon. In each event, $p_1$ points were given to the first place, $p_2$ points to the second place and $p_3$ points to third place, with $p_1> p_2> p_3> 0$ where $p_1$, $p_2$ and $p_3$ are integer numbers. The final result was $22$ points for $A$, $9$ for $B$, and $9$ for $C$. $B$ won the $100$ meter dash. Determine $M$ and who was the second in high jump.

John found all real numbers $p$ such that in the polynomial $g(x) = (x -1)^2(p + 2x)^2$ , the quadratic term has coefficient $2021$. What is the sum of all of these values $p$?

Two finite sequences $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ are just rearranged sequence $1, 1/2, ... , 1/n$ with $$a_1+b_1\ge a_2+b_2\ge...\ge a_n+b_n.$$ Prove that $a_m+a_n\ge 4/m$ for every $m$ ($1\le m\le n$) .

For pairwise distinct nonnegative reals $a,b,c$, prove that $$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$.

Let $N_0$ be the set of all non-negative integers. Let $f : N_0 \times N_0 \to N_0$ be a function such that for all non-negative integers $a,b$: $$f(a,b) = f(b,a),$$ $$f(a,0) = 0,$$ $$f(a+b,b) = f(a,b) +b.$$ Compute $$\sum_{i=0}^{30}\sum_{j=0}^{2^i-1}f(2^i, j)$$