Olympiad problems

2007 Today's Calculation Of Integral, 175

Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$

2009 AIME Problems, 7

Tags: induction , invariant , logarithm

The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$.

2017 Romania National Olympiad, 1

Tags: algebra , fractional part , integer part , floor , logarithm , equation

Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $ $ [] $ and $ \{\} $ [i]denote the floor, respectively, the fractional part.[/i]

2006 Putnam, A2

Tags: factorial , algebra , polynomial , function , probability , logarithm

Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)

2009 Today's Calculation Of Integral, 516

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

2012 Today's Calculation Of Integral, 802

Tags: calculus , integration , geometric transformation , rotation , ratio , logarithm , geometry

Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of the solid by a rotation of the figure enclosed by the curve $C$, the line $y=\frac{a}{a+k}$ and the $y$-axis around the $y$-axis. Find the ratio $\frac{V_2}{V_1}.$

2007 Stanford Mathematics Tournament, 15

Tags: integration , trigonometry , logarithm

Evaluate $\int_{0}^{\infty}\frac{\tan^{-1}(\pi x)-\tan^{-1}x}{x}dx$

2014 Saudi Arabia IMO TST, 2

Tags: function , floor function , induction , algebra , logarithm

Determine all functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(0)=0$ and \[f(x)=1+5f\left(\left\lfloor{\frac{x}{2}\right\rfloor}\right)-6f\left(\left\lfloor{\frac{x}{4}\right\rfloor}\right)\] for all $x>0$.

2007 Romania National Olympiad, 1

Tags: calculus , integration , inequalities , function , derivative , trigonometry , logarithm

Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained. [hide="Comment"] In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''. [/hide]

2010 AIME Problems, 5

Tags: sfft , special factorizations , logarithm

Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.

2005 Putnam, B3

Tags: function , integration , algebra , functional equation , absolute value , logarithm

Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that \[ f'\left(\frac ax\right)=\frac x{f(x)} \] for all $x>0.$

2006 AMC 12/AHSME, 20

Tags: probability , floor function , function , calculus , integration , inequalities , logarithm

Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that \[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? \]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$. $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$

2009 Unirea, 4

Tags: limit , trigonometry , geometry , modular arithmetic , probability , integration , logarithm

Evaluate the limit: \[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\] Proposed to "Unirea" Intercounty contest, grade 11, Romania

1981 AMC 12/AHSME, 15

Tags: logarithm

If $b>1$, $x>0$ and $(2x)^{\log_b 2}-(3x)^{\log_b 3}=0$, then $x$ is $\text{(A)}\ \frac{1}{216} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ \text{not uniquely determined}$

2011 Math Prize For Girls Problems, 12

Tags: floor function , logarithm

If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by \[ S(n) = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor + 10 \left( n - 10^{\lfloor \log n \rfloor} \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor \right) \, . \] (All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?

2005 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , limit , logarithm

Calculate \[ \lim_{x \to 0^+} \left( x^{x^x} - x^x \right). \]

1976 AMC 12/AHSME, 20

Tags: quadratic , logarithm

Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\] $\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$ $\textbf{(B) }\text{if and only if }a=b^2\qquad$ $\textbf{(C) }\text{if and only if }b=a^2\qquad$ $\textbf{(D) }\text{if and only if }x=ab\qquad$ $ \textbf{(E) }\text{for none of these}$

1965 AMC 12/AHSME, 6

Tags: logarithm

If $ 10^{\log_{10}9} \equal{} 8x \plus{} 5$ then $ x$ equals: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {5}{8} \qquad \textbf{(D)}\ \frac {9}{8} \qquad \textbf{(E)}\ \frac {2\log_{10}3 \minus{} 5}{8}$

2002 AMC 12/AHSME, 22

Tags: logarithm

For all integers $ n$ greater than $ 1$, define $ a_n \equal{} \frac {1}{\log_n 2002}$. Let $ b \equal{} a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5$ and $ c \equal{} a_{10} \plus{} a_{11} \plus{} a_{12} \plus{} a_{13} \plus{} a_{14}$. Then $ b \minus{} c$ equals $ \textbf{(A)}\ \minus{} 2 \qquad \textbf{(B)}\ \minus{} 1 \qquad \textbf{(C)}\ \frac {1}{2002} \qquad \textbf{(D)}\ \frac {1}{1001} \qquad \textbf{(E)}\ \frac {1}{2}$

2010 Today's Calculation Of Integral, 530

Tags: calculus , integration , geometry , calculus computations , logarithm

Answer the following questions. (1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$. (2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$. (3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.

2009 Today's Calculation Of Integral, 457

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$

2005 Today's Calculation Of Integral, 4

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Calculate the following indefinite integrals. [1] $\int \frac{x}{\sqrt{5-x}}dx$ [2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$ [3] $\int (\sin x+\cos x)^2dx$ [4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$ [5]$\int (\sin x+\sin 2x)^2 dx$

2010 Today's Calculation Of Integral, 636

Tags: calculus , integration , limit , derivative , calculus computations , logarithm , geometry

Let $a>1$ be a constant. In the $xy$-plane, let $A(a,\ 0),\ B(a,\ \ln a)$ and $C$ be the intersection point of the curve $y=\ln x$ and the $x$-axis. Denote by $S_1$ the area of the part bounded by the $x$-axis, the segment $BA$ and the curve $y=\ln x$ (1) For $1\leq b\leq a$, let $D(b,\ \ln b)$. Find the value of $b$ such that the area of quadrilateral $ABDC$ is the closest to $S_1$ and find the area $S_2$. (2) Find $\lim_{a\rightarrow \infty} \frac{S_2}{S_1}$. [i]1992 Tokyo University entrance exam/Science[/i]

2012 IMC, 3

Tags: logarithm

Is the set of positive integers $n$ such that $n!+1$ divides $(2012n)!$ finite or infinite? [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

2012 IberoAmerican, 3

Tags: floor function , number theory , logarithm

Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits. (Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)