Olympiad problems

2006 Harvard-MIT Mathematics Tournament, 10

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 $g(f(4))$.

1949 Putnam, B2

Tags: divergence , logarithm

Answer either (i) or (ii): (i) Prove that $$\sum_{n=2}^{\infty} \frac{\cos (\log \log n)}{\log n}$$ diverges. (ii) Assume that $p>0, a>0$, and $ac-b^{2} >0,$ and show that $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{ dx\; dy}{(p+ax^2 +2bxy+ cy^2 )^{2}}= \pi p^{-1} (ac-b^{2})^{- 1\slash 2}.$$

2011 Today's Calculation Of Integral, 728

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

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

PEN G Problems, 20

Tags: irrational number , floor function , logarithm

You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.

2006 Vietnam National Olympiad, 1

Tags: function , system of equations , algebra , logarithm

Solve the following system of equations in real numbers: \[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]

2008 Moldova MO 11-12, 8

Tags: integration , trigonometry , logarithm , real analysis

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

2004 South East Mathematical Olympiad, 3

Tags: polynomial , inequalities , algebra , logarithm

(1) Determine if there exists an infinite sequence $\{a_n\}$ with positive integer terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$. (2) Determine if there exists an infinite sequence $\{a_n\}$ with positive irrational terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.

2013 AMC 12/AHSME, 14

Tags: algebra , function , domain , ratio , arithmetic sequence , logarithm

The sequence \[\log_{12}{162},\, \log_{12}{x},\, \log_{12}{y},\, \log_{12}{z},\, \log_{12}{1250}\] is an arithmetic progression. What is $x$? $ \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}$

2002 AIME Problems, 6

Tags: quadratic , algebra , system of equations , logarithm

The solutions to the system of equations \begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*} are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$

2023 AMC 12/AHSME, 7

Tags: logarithm

For how many integers $n$ does the expression \[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}} \] represent a real number, where log denotes the base $10$ logarithm? $ \textbf{(A) }900 \qquad \textbf{(B) }2\qquad \textbf{(C) }902 \qquad \textbf{(D) } 2 \qquad \textbf{(E) }901$

1964 AMC 12/AHSME, 21

Tags: quadratic , logarithm

If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$, then $x$ equals: $ \textbf{(A)}\ 1/b^2 \qquad\textbf{(B)}\ 1/b \qquad\textbf{(C)}\ b^2 \qquad\textbf{(D)}\ b \qquad\textbf{(E)}\ \sqrt{b} $

2009 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , logarithm

Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]

2018 Moscow Mathematical Olympiad, 7

Tags: algebra , logarithm

$x^3+(\log_2{5}+\log_3{2}+\log_5{3})x=(\log_2{3}+\log_3{5}+\log_5{2})x^2+1$

2004 IMC, 6

Tags: function , polynomial , calculus , derivative , integration , logarithm , algebra

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

2004 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , limit , ratio , algebra , polynomial , logarithm , absolute value

Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.

2013 India National Olympiad, 6

Tags: algebra , polynomial , function , logarithm , inequalities

Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$

PEN E Problems, 14

Tags: algebra , polynomial , limit , function , gauss , number theory , logarithm

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

1949-56 Chisinau City MO, 40

Tags: algebra , system of equations , logarithm

Solve the system of equations: $$\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z =2 \\ \log_{3} y + \log_{9} z + \log_{9} x =2 \\ \log_{4} z + \log_{16} x + \log_{16} y =2\end{cases}$$

2005 Today's Calculation Of Integral, 3

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

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

2010 Today's Calculation Of Integral, 646

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

Evaluate \[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\] Own

2006 Hong kong National Olympiad, 2

Tags: function , modular arithmetic , algebra , logarithm

For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1}$ = $f_1 \circ f_n$ . Evaluate $f_{2007}(2^{2006} )$.

2013 Stanford Mathematics Tournament, 8

Tags: logarithm

According to Moor's Law, the number of shoes in Moor's room doubles every year. In 2013, Moor's room starts out having exactly one pair of shoes. If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes? Note that left and right shoes are distinct, and Moor must always wear one of each.

1948 Moscow Mathematical Olympiad, 145

Tags: logarithm , inequalities , algebra

Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$

2014 Contests, 1

Tags: ceiling function , combinatorics , logarithm

Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

2003 Brazil National Olympiad, 3

Tags: combinatorics , logarithm

A graph $G$ with $n$ vertices is called [i]cool[/i] if we can label each vertex with a different positive integer not greater than $\frac{n^2}{4}$ and find a set of non-negative integers $D$ so that there is an edge between two vertices iff the difference between their labels is in $D$. Show that if $n$ is sufficiently large we can always find a graph with $n$ vertices which is not cool.