Olympiad problems

2010 Contests, 1

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

2012 Stanford Mathematics Tournament, 10

Tags: logarithm , ceiling function , probability

Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)

2008 Moldova National Olympiad, 12.8

Tags: trigonometry , integration , real analysis , logarithm

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

2005 France Team Selection Test, 4

Tags: floor function , number theory , induction , logarithm , inequalities , ceiling function

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

2007 Ukraine Team Selection Test, 3

Tags: logarithm , algebra , polynomial

It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.

2009 Today's Calculation Of Integral, 461

Tags: calculus computations , inequalities , calculus , integration , limit , logarithm

Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$. (1) Find $ I_1,\ I_2$. (2) Find $ \lim_{n\to\infty} I_n$.

1998 South africa National Olympiad, 1

Tags: prime factorization , rational number , irrational number , logarithm , number theory

Is $\log_{10}{8}$ rational?

1972 AMC 12/AHSME, 29

Tags: logarithm , function

If $f(x)=\log \left(\frac{1+x}{1-x}\right)$ for $-1<x<1$, then $f\left(\frac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$ is $\textbf{(A) }-f(x)\qquad\textbf{(B) }2f(x)\qquad\textbf{(C) }3f(x)\qquad$ $\textbf{(D) }\left[f(x)\right]^2\qquad \textbf{(E) }[f(x)]^3-f(x)$

1988 Iran MO (2nd round), 3

Tags: logarithm , number theory

Let $n$ be a positive integer. $1369^n$ positive rational numbers are given with this property: if we remove one of the numbers, then we can divide remain numbers into $1368$ sets with equal number of elements such that the product of the numbers of the sets be equal. Prove that all of the numbers are equal.

2012 China Team Selection Test, 1

Tags: number theory , floor function , function , logarithm

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2012 Iran MO (3rd Round), 3

Tags: induction , function , logarithm , number theory , ceiling function

Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$. [i]Proposed by Amirhossein Gorzi[/i]

2016 District Olympiad, 1

Tags: algebra , equation , logarithm

Solve in the interval $ (2,\infty ) $ the following equation: $$ 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) . $$

2015 AMC 12/AHSME, 14

Tags: logarithm

What is the value of $a$ for which $\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1$? $\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad\textbf{(E) }36$

1970 IMO Longlists, 29

Tags: logarithm , number theory

Prove that the equation $4^x +6^x =9^x$ has no rational solutions.

2010 Today's Calculation Of Integral, 601

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

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

2003 Purple Comet Problems, 6

Tags: logarithm

Evaluate: \[\frac{1}{\log_2 (\frac{1}{6})} - \frac{1}{\log_3 (\frac{1}{6})} - \frac{1}{\log_4 (\frac{1}{6})}\]

2010 Today's Calculation Of Integral, 638

Tags: logarithm , integration , rotation , function , calculus , calculus computations

Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$ 1978 Chuo university entrance exam/Science and Technology

2012 Putnam, 4

Tags: calculus , inequalities , integration , induction , limit , logarithm

Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)

2009 Moldova Team Selection Test, 2

Tags: calculus , derivative , inequalities , rearrangement inequality , function , logarithm

[color=darkred]Let $ m,n\in \mathbb{N}$, $ n\ge 2$ and numbers $ a_i > 0$, $ i \equal{} \overline{1,n}$, such that $ \sum a_i \equal{} 1$. Prove that $ \small{\dfrac{a_1^{2 \minus{} m} \plus{} a_2 \plus{} ... \plus{} a_{n \minus{} 1}}{1 \minus{} a_1} \plus{} \dfrac{a_2^{2 \minus{} m} \plus{} a_3 \plus{} ... \plus{} a_n}{1 \minus{} a_1} \plus{} ... \plus{} \dfrac{a_n^{2 \minus{} m} \plus{} a_1 \plus{} ... \plus{} a_{n \minus{} 2}}{1 \minus{} a_1}\ge n \plus{} \dfrac{n^m \minus{} n}{n \minus{} 1}}$[/color]

2007 Princeton University Math Competition, 6

Tags: logarithm , probability , integration , calculus , geometry , trapezoid

Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

2010 Today's Calculation Of Integral, 522

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

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

Today's calculation of integrals, 898

Tags: calculus , integration , logarithm , calculus computations

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

2012 Today's Calculation Of Integral, 788

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

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

1984 AIME Problems, 5

Tags: logarithm , symmetry

Determine the value of $ab$ if $\log_8 a + \log_4 b^2 = 5$ and $\log_8 b + \log_4 a^2 = 7$.

Today's calculation of integrals, 885

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

Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$