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

Two triangles $ABC$ and $A'B'C'$ are on the plane. It is known that each side length of triangle $ABC$ is not less than $a$, and each side length of triangle $A'B'C'$ is not less than $a'$. Prove that we can always choose two points in the two triangles respectively such that the distance between them is not less than $\sqrt{\dfrac{a^2+a'^2}{3}}$.

Let $n\in\mathbb{N}, x_0=0$ and $x_1,x_2,\ldots,x_n$ be postive real numbers such that $x_1+x_2+\ldots+x_n=1$. Show that $$1\leq\sum_{i=1}^{n}\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}<\frac{\pi}{2}.$$

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $$\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1.$$

The sequences $(a_n), (b_n),$ and $(c_n)$ are defined by $a_0 = 1, b_0 = 0, c_0 = 0,$ and \[a_n = a_{n-1} + \frac{c_{n-1}}{n}, b_n = b_{n-1} +\frac{a_{n-1}}{n}, c_n = c_{n-1} +\frac{b_{n-1}}{n}\] for all $n \geq1$. Prove that \[\left|a_n -\frac{n + 1}{3}\right|<\frac{2}{\sqrt{3n}}\] for all $n \geq 1$.

Given $a,b,c \in \mathbb{R}$ satisfying $a+b+c=a^2+b^2+c^2=1$, show that $\frac{-1}{4} \leq ab \leq \frac{4}{9}$.

Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

Given nonnegative real numbers $x_1,x_2,\dots ,x_n$, let $a=min\{x_1, x_2,\dots ,x_n\}$. Prove that the following inequality holds: \[ \sum^{n}_{i=1}\dfrac{1+x_i}{1+x_{i+1}}\le n+\dfrac{1}{(1+a)^2}\sum^{n}_{i=1}(x_i-a)^2 \quad\quad (x_{n+1}=x_1),\] and equality occurs if and only if $x_1=x_2=\dots =x_n$.

The incircle of a right triangle $ABC$ touches its hypotenuse $BC$ at point $D$. The line $AD$ intersects the circumscribed circle at point $X$. Prove that $ |BX-CX| \geqslant |AD - DX|$.

(10.4) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $0 \le a_{k+1}- a_k \le 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \le \left(\sum_{k=1}^n a_k \right)^2$$ (11.3) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $a_{k+1} \ge a_k + 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \ge \left(\sum_{k=1}^n a_k \right)^2$$

Let $x,y,z$ be non-negative real numbers. Prove that: \[\sqrt{(2x+y)(2x+z)}+\sqrt{(2y+x)(2y+z)}+\sqrt{(2z+x)(2z+y)}\geq \] \[\geq \sqrt{(x+2y)(x+2z)}+\sqrt{(y+2x)(y+2z)}+\sqrt{(z+2x)(z+2y)}.\]

Prove that if $a, b, c$ are positive numbers and $ab + bc + ca > a+ b + c$, then $a + b + c > 3$.

Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

Let $ a,b,c$ be positive reals; show that $ \displaystyle a \plus{} b \plus{} c \leq \frac {bc}{b \plus{} c} \plus{} \frac {ca}{c \plus{} a} \plus{} \frac {ab}{a \plus{} b} \plus{} \frac {1}{2}\left(\frac {bc}{a} \plus{} \frac {ca}{b} \plus{} \frac {ab}{c}\right)$ [i](Darij Grinberg)[/i]

Given positive reals $x,y,z$ satisfying $x+y+z=3$, prove that \[\sum_{cyc}\left( x^2+y^2+x^2y^2+\frac{y^2}{x^2}\right)\geq 4\sum_{cyc}\frac{y}{x}.\] [i]Proposed by chengbilly.[/i]

Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\] Prove that $P(x)$ do not have a real root in $[-1,1]$.

Let $W,A,B$ be fixed real numbers with $W>0$. Prove that the following statements are equivalent. [list] [*] For all $x, y, z\ge 0$ satisfying $x+y\le z+W, x+z\le y+W, y+z\le x+W$ we have $Axyz+B\ge x^2+y^2+z^2$. [*] $B\ge W^2$ and $AW^3+B\ge 3W^2$. [/list] [i]Proposed by Ákos Somogyi, London[/i]

Let $ x$, $ y$ and $ z$ be non negative numbers. Prove that \[ \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}\]

Determine the maximal value of $ k $, such that for positive reals $ a,b $ and $ c $ from inequality $ kabc >a^3+b^3+c^3 $ it follows that $ a,b $ and $ c $ are sides of a triangle.

What is the largest real number $C$ that satisfies the inequality $x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)$ for every real $x$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 25 $

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

Let $a, b, c$ be the side lengths of a triangle. Prove that $2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)$

$a,b,c$ are real numbers. The sufficient and necessary condition of $\forall x\in\mathbb{R},a\sin x+b\cos x+c>0$ is $\text{(A)}$ $a=b=0,c>0$ $\text{(B)}$ $\sqrt{a^2+b^2}=c$ $\text{(C)}$ $\sqrt{a^2+b^2}<c$ $\text{(D)}$ $\sqrt{a^2+b^2}>c$