Olympiad problems

2007 Today's Calculation Of Integral, 225

Tags: calculus , integration , geometry , ratio , inequalities , function , logarithm

2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.

1985 Canada National Olympiad, 4

Tags: floor function , inequalities , number theory , prime factorization , logarithm

Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.

2001 Junior Balkan Team Selection Tests - Romania, 1

Tags: function , geometry , logarithm

Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$.

1997 Finnish National High School Mathematics Competition, 1

Tags: parameterization , quadratic , polynomial , product of roots , algebra , logarithm

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

2011 Postal Coaching, 5

Tags: logarithm , inequalities

Let $<a_n>$ be a sequence of non-negative real numbers such that $a_{m+n} \le a_m +a_n$ for all $m,n \in \mathbb{N}$. Prove that \[\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N\] for any $N \in \mathbb{N}$, where $\ln$ denotes the natural logarithm.

2009 China Second Round Olympiad, 2

Tags: inequalities , logarithm

Let $n$ be a positive integer. Prove that \[-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}\]

2003 Moldova National Olympiad, 12.1

Tags: limit , calculus , calculus computations , logarithm

For every natural number $n$ let: $a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find: \[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].

2012 Today's Calculation Of Integral, 821

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

Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$

1996 Putnam, 2

Tags: inequalities , induction , geometry , rectangle , integration , logarithm

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

2006 Moldova National Olympiad, 10.5

Tags: inequalities , calculus , logarithm , derivative , function , algebra

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

1998 USAMTS Problems, 1

Tags: logarithm

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

2009 Indonesia TST, 1

Tags: induction , inequalities , combinatorics , logarithm

2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

Today's calculation of integrals, 897

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

Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

2007 Princeton University Math Competition, 5

Tags: logarithm

Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.

2022 Romania National Olympiad, P1

Tags: algebra , logarithm

Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$ [i]Mihai Opincariu[/i]

2007 AIME Problems, 12

Tags: calculus , integration , ratio , logarithm

The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that \[\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,\] find $\log_{3}(x_{14}).$

2014 Contests, 1

Tags: limit , induction , logarithm

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 $\displaystyle\lim_{n\to\infty} a_n =0$.

2007 Today's Calculation Of Integral, 225

1985 Canada National Olympiad, 4

2001 Junior Balkan Team Selection Tests - Romania, 1

1997 Finnish National High School Mathematics Competition, 1

2011 Postal Coaching, 5

2009 China Second Round Olympiad, 2

2003 Moldova National Olympiad, 12.1

2012 Today's Calculation Of Integral, 821

1996 Putnam, 2

2006 Moldova National Olympiad, 10.5

1998 USAMTS Problems, 1

2009 Indonesia TST, 1

Today's calculation of integrals, 897

2007 Princeton University Math Competition, 5

2022 Romania National Olympiad, P1

2007 AIME Problems, 12

2014 Contests, 1

2009 Today's Calculation Of Integral, 448

2002 District Olympiad, 3

2020 AMC 12/AHSME, 10

2005 Today's Calculation Of Integral, 82

1956 AMC 12/AHSME, 18

2012 ELMO Shortlist, 2

1973 Swedish Mathematical Competition, 1

PEN D Problems, 22