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 all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

Let $a_1,a_2,\ldots$ be a sequence defined by $a_1=a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for $n\geq 1$. Find \[\sum_{n=1}^\infty \dfrac{a_n}{4^{n+1}}.\]

Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function $($that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}})$ for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous. [i]Proposed by Zoltán Boros[/i]

Let $ p>2 $ be a prime number. For any permutation $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $ of the set $ S = \{ 1, 2, \cdots , p \} $, let $ f( \pi ) $ denote the number of multiples of $ p $ among the following $ p $ numbers: \[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) \] Determine the average value of $ f( \pi) $ taken over all permutations $ \pi $ of $ S $.

Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$. $\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$. $\text{b})$ Prove that the amount of the limit does [b][u]not[/u][/b] depend on choosing $x$.

Let be two real numbers $ a<b $ and a differentiable function $ f:[a,b]\longrightarrow\mathbb{R} $ that has a bounded derivative. Show that if $ \frac{f(b)-f(a)}{b-a} $ is equal to the global supremum or infimum of $ f', $ then $ f $ is polynomial with degree $ 1. $ [i]Cătălin Țigăeru[/i]

Find the number of ordered triplets of nonnegative integers $(m, n, p)$ such that $m+3n+5p \le 600$.

Find all functions $f: \mathbb{Q}[x] \to \mathbb{R}$ such that: (a) for all $P, Q \in \mathbb{Q}[x]$, $f(P \circ Q) = f(Q \circ P);$ (b) for all $P, Q \in \mathbb{Q}[x]$ with $PQ \neq 0$, $f(P\cdot Q) = f(P) + f(Q).$ ($P \circ Q$ indicates $P(Q(x))$.)

Let $ x$,$ y$, and $ z$ be three positive real numbers whose sum is $ 1$. If no one of these numbers is more than twice any other, then the minimum possible value of the product $ xyz$ is $ \textbf{(A)}\ \frac{1}{32}\qquad \textbf{(B)}\ \frac{1}{36}\qquad \textbf{(C)}\ \frac{4}{125}\qquad \textbf{(D)}\ \frac{1}{127}\qquad \textbf{(E)}\ \text{none of these}$

How many positive integer multiples of 1001 can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0\leq i < j \leq 99$?

Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[f(x+y)=f(x-y)+f(f(1-xy))\] holds for all real numbers $x$ and $y$

Let $\{a_n\}_{n\geq 0}$ be an arithmetic sequence with difference $d$ and $1\leq a_0\leq d$. Denote the sequence as $S_0$, and define $S_n$ recursively by two operations below: Step $1$: Denote the first number of $S_n$ as $b_n$, and remove $b_n$. Step $2$: Add $1$ to the first $b_n$ numbers to get $S_{n+1}$. Prove that there exists a constant $c$ such that $b_n=[ca_n]$ for all $n\geq 0$, where $[]$ is the floor function.

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$ Find the value of $\frac{I_1}{I_2}.$

Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that $$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$

Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$

We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\

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 $x \in \mathbb{R}$

Let $S=\{1,2,\cdots ,n\}$ where $n$ is an odd integer. Let $f$ be a function defined on $\{(i,j): i\in S, j \in S\}$ taking values in $S$ such that (i) $f(s,r)=f(r,s)$ for all $r,s \in S$ (ii) $\{f(r,s): s\in S\}=S$ for all $r\in S$ Show that $\{f(r,r): r\in S\}=S$

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

Let $S$ be the set of all non-increasing sequences of numbers $a_1 \geq a_2 \geq \ldots \geq a_{101}$ such that $a_i \in \{ 0,1,\ldots ,101 \}$ for all $1 \leq i \leq 101$ For every sequence $s \in S$ let $$f(s)=\lceil \frac{a_1}{2} \rceil+\lfloor \frac{a_2}{2} \rfloor + \lceil \frac{a_3}{2} \rceil + \ldots + \lfloor \frac{a_{100}}{2} \rfloor + \lceil \frac{a_{101}}{2} \rceil$$ where $\lfloor x \rfloor$ is the greatest integer, not exceeding $x$, and $\lceil x \rceil$ is the least integer at least $x$. Prove that the number of sequences $s \in S$ for which $f(s)$ is even is the same, as the number of sequences $s$ for which $f(s)$ is odd [i]M. Zorka[/i]

Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, \[f(xf(y))+f(y)=f(x+y)+f(xy).\] [i]Milan Haiman[/i]

Find all functions $h: \mathbb{Z}\to \mathbb{Z}$ such that for all $x,y\in \mathbb{Z}$: \[h(x+y)+h(xy)=h(x)h(y)+1.\]

A function $g$ defined for all positive integers $n$ satisfies [list][*]$g(1) = 1$; [*]for all $n\ge 1$, either $g(n+1)=g(n)+1$ or $g(n+1)=g(n)-1$; [*]for all $n\ge 1$, $g(3n) = g(n)$; and [*]$g(k)=2001$ for some positive integer $k$.[/list] Find, with proof, the smallest possible value of $k$.