Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds: $$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$

If $a > b > c > d > 0$ are integers such that $ad = bc$, show that $$(a - d)^2 \ge 4d + 8$$

Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$. Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.

Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B' + B' C' + C' A' \le DA + DB + DC$. (V.Yassinsky)

For positive $a,b,c$, such that $abc=1$ prove the inequality: $4(\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}) \leq 3(2+a+b+c+\frac{1}{a}+\frac{1}{b}+ \frac{1}{c})^{\frac{2}{3}}$.

For each integer $n \ge 1$, prove that there is a polynomial $P_{n}(x)$ with rational coefficients such that $x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}$. Define the rational number $a_{n}$ by \[a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.\] Prove that $a_{n}$ satisfies the inequality \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]

a) Prove that if $x,y>0$ then \[ \frac x{y^2} + \frac y{x^2} \geq \frac 1x + \frac 1y. \] b) Prove that if $a,b,c$ are positive real numbers, then \[ \frac {a+b}{c^2} + \frac {b+c}{a^2} + \frac {c+a}{b^2} \geq 2 \left( \frac 1a + \frac 1b + \frac 1c \right). \]

Let $x,y,z>0$ such that $\frac1x+\frac1y+\frac1z<\frac{1}{xyz}$. Show that \[\frac{2x}{\sqrt{1+x^2}}+\frac{2y}{\sqrt{1+y^2}}+\frac{2z}{\sqrt{1+z^2}}<3.\]

A function \[\text{f:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] is called contract if, for every numbers $x,y\in \text{(0,}\infty \text{)}$ we have, $\underset{n\to \infty }{\mathop{\lim }}\,\left( {{f}^{n}}\left( x \right)-{{f}^{n}}\left( y \right) \right)=0$ where ${{f}^{n}}=\underbrace{f\circ f\circ ...\circ f}_{n\ f\text{'s}}$ a) Consider \[f:\text{(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] a function contract, continue with the property that has a fixed point, that existing ${{x}_{0}}\in \text{(0,}\infty \text{) }$ there so that $f\left( {{x}_{0}} \right)={{x}_{0}}.$ Show that $f\left( x \right)>x,$ for every $x\in \text{(0,}{{x}_{0}}\text{)}\,$ and $f\left( x \right)<x$, for every $x\in \text{(}{{x}_{0}}\text{,}\infty \text{)}\,$. b) Show that the given function \[f\text{:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] given by $f\left( x \right)=x+\frac{1}{x}$ is contracted but has no fix number.

Show that for a triangle we have \[ \max \{am_a,bm_b,cm_c\} \leq sR \] where $m_a$ denotes the length of median of side $BC$ and $s$ is half of the perimeter of the triangle.

A ship sails $ 10$ miles in a straight line from $ A$ to $ B$, turns through an angle between $ 45^{\circ}$ and $ 60^{\circ}$, and then sails another $ 20$ miles to $ C$. Let $ AC$ be measured in miles. Which of the following intervals contains $ AC^2$? [asy]unitsize(2mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair B=(0,0), A=(-10,0), C=20*dir(50); draw(A--B--C); draw(A--C,linetype("4 4")); dot(A); dot(B); dot(C); label("$10$",midpoint(A--B),S); label("$20$",midpoint(B--C),SE); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE);[/asy]$ \textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800]$ $ \textbf{(E)}\ [800,900]$

Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds: \[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\] where $S=\sum_{i=1}^n x_i$.

For positive numbers $ x_1, x_2, \ldots, x_n $ $ (n \geq 1) $ the following equality holds $$ \frac {1} {{1 + x_1}} + \frac {1} {{1 + x_2}} + \ldots + \frac {1} {{1 + x_n}} = 1. $$ Prove that $ x_1 \cdot x_2 \cdot \ldots \cdot x_n \geq (n-1) ^ n. $

Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.

Let $P_1 , P_2 , ...$ be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A , $\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1$. Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.

Let $a, b, c$ be positive real numbers satisfying the condition $\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}=1$ . Prove the inequality $$\frac{a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}} \le \frac{3\sqrt3}{2}$$ When does the equality hold? (L. Parts)

Let $ABC$ be a triangle such that \[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \] where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0, y_1+y_2+y_3=0$. Prove that \[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\] [i]Ray Li, Max Schindler.[/i]

Let $ m,n$ be two natural nonzero numbers and sets $ A \equal{} \{ 1,2,...,n\}, B \equal{} \{1,2,...,m\}$. We say that subset $ S$ of Cartesian product $ A \times B$ has property $ (j)$ if $ (a \minus{} x)(b \minus{} y)\le 0$ for each pairs $ (a,b),(x,y) \in S$. Prove that every set $ S$ with propery $ (j)$ has at most $ m \plus{} n \minus{} 1$ elements. [color=#FF0000]The statement was edited, in order to reflect the actual problem asked. The sign of the inequality was inadvertently reversed into $ (a \minus{} x)(b \minus{} y)\ge 0$, and that accounts for the following two posts.[/color]

Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which \[EG+3HF\ge kd+(1-k)s \] where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?

Let $S$ be the set of all $n$-tuples of real numbers, with the property that among the numbers $x_1,\frac{x_1+x_2}2,\ldots,\frac{x_1+x_2+\ldots+x_n}n$ the least is equal to $0$, and the greatest is equal to $1$. Determine $$\max_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j)\qquad\text{and}\min_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j).$$

Let $a, b, c$ be positive real numbers with $ab + bc + ac = abc$. Prove that $$\frac{bc}{a^{a+1}} +\frac{ac}{b^{b+1 }}+\frac{ab}{c^{c+1}} \ge \frac13$$

For a finite set $A$ of integers, define $s(A)$ as the number of values obtained by adding any two elements of $A$, not necessarily different. Analogously, define $r (A)$ as the number of values obtained by subtracting any two elements of $A$, not necessarily different. For example, if $A = \{3,1,-1\}$ $\bullet$ The values obtained by adding any two elements of $A$ are $\{6,4,2,0,-2\}$ and so $s (A) = 5$. $\bullet$ The values obtained by subtracting any two elements of $A$ are $\{4,2,0,-2,-4\}$ and as $r (A) = 5$. Prove that for each positive integer $n$ there is a finite set $A$ of integers such that $r (A) \ge n s (A)$.

Let $S$ denote the set of $2$ by $2$ matrices with integer entries and determinant $1$, and let $T$ denote those matrices of $S$ which are congruent to the identity matrix $I\pmod3$ (so $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in T$ means that $a,b,c,d\in\mathbb Z,ad-bc=1,$ and $3$ divides $b,c,a-1,d-1$). (a) Let $f:T\to\mathbb R$ be a function such that for every $X,Y\in T$ with $Y\ne I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$. Show that given two finite nonempty subsets $A,B$ of $T$, there are matrices $a\in A$ and $b\in B$ such that if $a'\in A$, $b'\in B$ and $a'b'=ab$, then $a'=a$ and $b'=b$. (b) Show that there is no $f:S\to\mathbb R$ such that for every $X,Y\in S$ with $Y\ne\pm I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$.