Given $a_0 > 1$, the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$, $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence. Find all $a_0$ for which all terms of the sequence are primes or prime powers.

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

Solve the equation \[(x+1)\log^2_{3}x+4x\log_{3}x-16=0\]

If $x, y, z, t > 1$ then: $$\left(\log _{zxt}x\right)^2+\left(\log _{xyt}y\right)^2+\left(\log _{xyz}z\right)^2+\left(\log _{yzt}t\right)^2>\frac{1}{4}$$

A line $ x \equal{} k$ intersects the graph of $ y \equal{} \log_5{x}$ and the graph of $ y \equal{} \log_5{(x \plus{} 4)}$. The distance between the points of intersection is $ 0.5$. Given that $ k \equal{} a \plus{} \sqrt{b}$, where $ a$ and $ b$ are integers, what is $ a \plus{} b$? $ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

prove that $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\pi}{4}$

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]

If $ \log_{10}{m} \equal{} b \minus{} \log_{10}{n}$, then $ m$= $\textbf{(A)}\ \dfrac{b}{n} \qquad \textbf{(B)}\ bn \qquad \textbf{(C)}\ 10^b n\qquad \textbf{(D)}\ b-10^n \qquad \textbf{(E)}\ \dfrac{10^b}{n}$

Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.

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 value of $\frac{\log_59\log_75\log_37}{\log_2\sqrt{6}}+\frac{1}{\log_9\sqrt{6}}$ $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }7 $

Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

Compute $\lim\limits_{n \to \infty} \frac{1}{\log \log n} \sum\limits_{k=1}^n (-1)^k \binom{n}{k} \log k.$

Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

Let $ a,b,c\in \mathbb R$ such that $ a\ge b\ge c > 1$. Prove the inequality: $ \log_c\log_c b \plus{} \log_b\log_b a \plus{} \log_a\log_a c\geq 0$

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

Evaluate $ \int_1^e \{(1 \plus{} x)e^x \plus{} (1 \minus{} x)e^{ \minus{} x}\}\ln x\ dx$.

Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.

Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.

What is the value of $(625^{\log_{5}{2015}})^{\frac{1}{4}}$? $\textbf{(A) }5\qquad\textbf{(B) }\sqrt[4]{2015}\qquad\textbf{(C) }625\qquad\textbf{(D) }2015\qquad\textbf{(E) }\sqrt[4]{5^{2015}}$

If $\log_2(\log_3(\log_4 x))=\log_3(\log_4(\log_2 y))=\log_4(\log_2(\log_3 z))=0$, then the sum $x+y+z$ is equal to $\textbf{(A) }50\qquad\textbf{(B) }58\qquad\textbf{(C) }89\qquad\textbf{(D) }111\qquad \textbf{(E) }1296$

Let $ p$ be a positive integer. Define the function $ f: \mathbb{N}\to\mathbb{N}$ by $ f(n)\equal{}a_1^p\plus{}a_2^p\plus{}\cdots\plus{}a_m^p$, where $ a_1, a_2,\ldots, a_m$ are the decimal digits of $ n$ ($ n\equal{}\overline{a_1a_2\ldots a_m}$). Prove that every sequence $ (b_k)^\infty_{k\equal{}0}$ of positive integer that satisfy $ b_{k\plus{}1}\equal{}f(b_k)$ for all $ k\in\mathbb{N}$, has a finite number of distinct terms. $ \mathbb{N}\equal{}\{1,2,3\ldots\}$

Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

If $a>1,b>1,\lg(a+b)=\lg a+\lg b$, then the value of $\lg(a-1)+\lg(b-1)$ is $\text{(A)}\lg2\qquad\text{(B)}1\qquad\text{(C)}0\qquad\text{(D)}$ not sure

The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$? $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$