Olympiad problems

2009 Today's Calculation Of Integral, 434

Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.

1998 USAMTS Problems, 1

Tags: USAMTS , logarithms

Determine the leftmost three digits of the number \[1^1+2^2+3^3+...+999^{999}+1000^{1000}.\]

2010 Today's Calculation Of Integral, 584

Tags: calculus , integration , limit , logarithms , calculus computations

Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

2011 Today's Calculation Of Integral, 748

Tags: calculus , integration , trigonometry , logarithms , calculus computations

Evaluate the following integrals. (1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$ (2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$

1997 National High School Mathematics League, 12

Tags: logarithms

Let $a=\lg z+\lg\left[x(yz)^{-1}+1\right],b=\lg x^{-1}+\lg(xyz+1),c=\lg y+\lg\left[(xyz)^{-1}+1\right]$, if $M=\max\{a,b,c\}$, then the minumum value of $M$ is________.

2011 Kosovo National Mathematical Olympiad, 3

Tags: inequalities , logarithms , inequalities proposed

Prove that the following inequality holds: \[ \left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4\]

2010 Contests, 522

Tags: calculus , integration , limit , logarithms , function , geometry , derivative

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

2008 Harvard-MIT Mathematics Tournament, 9

Tags: limit , integration , HMMT , logarithms , calculus , real analysis , calculus computations

([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

2008 Vietnam National Olympiad, 1

Tags: logarithms , calculus , limit , algebra unsolved , algebra

Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$

2005 USAMO, 6

Tags: logarithms , induction , modular arithmetic , ceiling function , floor function , pigeonhole principle , inequalities

For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with \[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]

Today's calculation of integrals, 871

Tags: calculus , integration , trigonometry , limit , logarithms , calculus computations , Definite integral

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

1983 AMC 12/AHSME, 12

Tags: logarithms

If $\log_7 \Big(\log_3 (\log_2 x) \Big) = 0$, then $x^{-1/2}$ equals $\displaystyle \text{(A)} \ \frac{1}{3} \qquad \text{(B)} \ \frac{1}{2 \sqrt 3} \qquad \text{(C)} \ \frac{1}{3 \sqrt 3} \qquad \text{(D)} \ \frac{1}{\sqrt{42}} \qquad \text{(E)} \ \text{none of these}$

2020 AIME Problems, 2

Tags: 2020 AIME I , logarithms , geometric sequence

There is a unique positive real number $x$ such that the three numbers $\log_8(2x),\log_4x,$ and $\log_2x,$ in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2003 District Olympiad, 4

Tags: logarithms , function , algebra unsolved , algebra

Let $\displaystyle a,b,c,d \in \mathbb R$ such that $\displaystyle a>c>d>b>1$ and $\displaystyle ab>cd$. Prove that $\displaystyle f : \left[ 0,\infty \right) \to \mathbb R$, defined through \[ \displaystyle f(x) = a^x+b^x-c^x-d^x, \, \forall x \geq 0 , \] is strictly increasing.

2018 AMC 12/AHSME, 14

Tags: AMC , AMC 12 , AMC 12 A , number theory , logarithms

The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\tfrac{1}{3}$ or $\tfrac{1}{2}$, can be written as $\tfrac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$? $\textbf{(A) } 5 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 31 \qquad \textbf{(E) } 35 $

2011 Today's Calculation Of Integral, 728

Tags: calculus , integration , trigonometry , logarithms , calculus computations

Evaluate \[\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.\]

2010 Today's Calculation Of Integral, 619

Tags: calculus , integration , function , trigonometry , logarithms , calculus computations

Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized. Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$ [i]2010 Saitama University entrance exam/Mathematics[/i] Last Edited

2018 India PRMO, 14

Tags: trigonometry , logarithms

If $x = cos 1^o cos 2^o cos 3^o...cos 89^o$ and $y = cos 2^o cos 6^o cos 10^o...cos 86^o$, then what is the integer nearest to $\frac27 \log_2 \frac{y}{x}$ ?

1954 Czech and Slovak Olympiad III A, 3

Tags: logarithm , inequalities , logarithms

Show that $$\log_2\pi+\log_4\pi<\frac52.$$

1978 IMO Longlists, 33

Tags: function , logarithms , inequalities , algebra proposed , algebra

A sequence $(a_n)^{\infty}_0$ of real numbers is called [i]convex[/i] if $2a_n\le a_{n-1}+a_{n+1}$ for all positive integers $n$. Let $(b_n)^{\infty}_0$ be a sequence of positive numbers and assume that the sequence $(\alpha^nb_n)^{\infty}_0$ is convex for any choice of $\alpha > 0$. Prove that the sequence $(\log b_n)^{\infty}_0$ is convex.

1972 AMC 12/AHSME, 8

Tags: logarithms , AMC

If $|x-\log y|=x+\log y$ where $x$ and $\log y$ are real, then $\textbf{(A) }x=0\qquad\textbf{(B) }y=1\qquad\textbf{(C) }x=0\text{ and }y=1\qquad$ $\textbf{(D) }x(y-1)=0\qquad \textbf{(E) }\text{None of these}$

2014 Dutch BxMO/EGMO TST, 5

Tags: ceiling function , logarithms , combinatorics proposed , combinatorics , greedy algorithm

Let $n$ be a positive integer. Daniel and Merlijn are playing a game. Daniel has $k$ sheets of paper lying next to each other on a table, where $k$ is a positive integer. On each of the sheets, he writes some of the numbers from $1$ up to $n$ (he is allowed to write no number at all, or all numbers). On the back of each of the sheets, he writes down the remaining numbers. Once Daniel is ﬁnished, Merlijn can ﬂip some of the sheets of paper (he is allowed to ﬂip no sheet at all, or all sheets). If Merlijn succeeds in making all of the numbers from $1$ up to n visible at least once, then he wins. Determine the smallest $k$ for which Merlijn can always win, regardless of Daniel’s actions.

1981 National High School Mathematics League, 8

Tags: logarithms

In the logarithm table below, there are two mistakes. Correct them. \begin{tabular}{|c|c|} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $\lg0.021$&$2a+b+c-3$ \\ \hline $\lg0.27$&$6a-3b-2$\\ \hline $\lg1.5$&$3a-b+c$\\ \hline $\lg2.8$&$1-2a+2b-c$\\ \hline $\lg3$&$2a-b$\\ \hline $\lg5$&$a+c$\\ \hline $\lg6$&$1+a-b-c$\\ \hline $\lg7$&$2(a+c)$\\ \hline $\lg8$&$3-3a-3c$\\ \hline $\lg9$&$4a-2b$\\ \hline $\lg14$&$1-a+2b$\\ \hline \end{tabular}

2017 District Olympiad, 2

Tags: System , equations , logarithms

Solve in $ \mathbb{Z} $ the system: $$ \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. . $$

2006 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function , integration , logarithms

Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.