For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$

Let $ x$, $ y$ and $ z$ be non negative reals, such that there are not two simultaneously equal to $ 0$. Show that $ \frac {x \plus{} y}{y \plus{} z} \plus{} \frac {y \plus{} z}{x \plus{} y} \plus{} \frac {y \plus{} z}{z \plus{} x} \plus{} \frac {z \plus{} x}{y \plus{} z} \plus{} \frac {z \plus{} x}{x \plus{} y} \plus{} \frac {x \plus{} y}{z \plus{} x}\geq\ 5 \plus{} \frac {x^{2} \plus{} y^{2} \plus{} z^{2}}{xy \plus{} yz \plus{} zx}$ and determine the equality cases.

Show that if $t_1 , t_2, t_3, t_4, t_5$ are real numbers, then $$\sum_{j=1}^{5} (1-t_j )\exp \left( \sum_{k=1}^{j} t_k \right) \leq e^{e^{e^{e}}}.$$

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$.

Positive numbers $x_1,..., x_k$ satisfy the following inequalities: $$x_1^2+...+ x_k^2 <\frac{x_1+...+x_k}{2} \ \ and \ \ x_1+...+x_k < \frac{x_1^3+...+ x_k^3}{2}$$ a) Show that $k > 50$, (3) b) Give an example of such numbers for some value of $k$ (3) c) Find minimum $k$, for which such an example exists. (3)

Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that $$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$

Let $ a_i, b_i$ be positive real numbers, $ i\equal{}1,2,\ldots,n$, $ n\geq 2$, such that $ a_i<b_i$, for all $ i$, and also \[ b_1\plus{}b_2\plus{}\cdots \plus{} b_n < 1 \plus{} a_1\plus{}\cdots \plus{} a_n.\] Prove that there exists a $ c\in\mathbb R$ such that for all $ i\equal{}1,2,\ldots,n$, and $ k\in\mathbb Z$ we have \[ (a_i\plus{}c\plus{}k)(b_i\plus{}c\plus{}k) > 0.\]

Let $ n $ be a positive integer and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $ a_1^2+a_2^2+\cdots +a_n^2=1. $ Show that $$ \sum_{1\le ij\le n} a_ia_j<2\sqrt n. $$ [i]Russian math competition[/i]

Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form \[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\] with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.

Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$

Let $a_1, a_2, a_3, ...$ be a sequence of positive real numbers. Prove that for any positive integer $n$ the inequality holds $\sum_{i=1}^n b_i^2 \le 4 \sum_{i=1}^n a_i^2$ where $b_i$ is the arithmetic mean of the numbers $a_1, a_2, ..., a_n$

Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that $$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$

Let $(a_n), n = 0, 1, . . .,$ be a sequence of real numbers such that $a_0 = 0$ and \[a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots\] Prove that there exists a positive number $q, q < 1$, such that for all $n = 1, 2, \ldots ,$ \[|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,\] and give one such $q$ explicitly.

Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$

Consider an infinite strictly increasing sequence of positive integers $a_1$, $a_2$,$...$ where for any real number $C$, there exists an integer $N$ where $a_k >Ck$ for any $k >N$. Do there necessarily exist inifinite many indices $k$ where $2a_k <a_{k-1}+a_{k+1}$ for any $0<i<k$?

If $ABC$ is acute triangle, prove distance from each vertex to corresponding excentre is less than sum of two greatest side of triangle

Suppose $n$ is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least $n$. Express the $n-$th smallest number in this set in terms of $n$.

Let $abc$ be real numbers satisfying $ab+bc+ca=1$. Show that $\frac{|a-b|}{|1+c^2|}$ + $\frac{|b-c|}{|1+a^2|}$ $>=$ $\frac{|c-a|}{|1+b^2|}$

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?

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

Find the set of solutions for $x$ in the inequality $\dfrac{x+1}{x+2}>\dfrac{3x+4}{2x+9}$ when $x\neq 2$, $x\neq -\dfrac{9}{2}$.

For non-negative numbers $x$ and $y$ not exceeding $1$, prove that $$\frac{1}{\sqrt{1+x^2}}+\frac{1}{\sqrt{1+y^2}} \le \frac{2}{\sqrt{1 + xy}},$$

An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1]. A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur. (1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

If $a^2+b^2+c^2+d^2=1$, prove that $$(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2\leq 4$$ When does equality holds?

On the domain $0\leq \theta \leq 2\pi:$ (a) Prove that $\sin^{2}\theta \cdot \sin 2\theta$ takes its maximum at $\frac{\pi}{3}$ and $\frac{4 \pi}{3}$ (and hence its minimum at $\frac{2 \pi}{3}$ and $\frac{ 5 \pi}{3}$). (b) Show that $$| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |$$ takes its maximum at $\frac{4 \pi}{3}$ (the maximum may also be attained at other points). (c) Derive the inequality: $$ \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.$$