Olympiad problems

2003 District Olympiad, 4

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.

1993 AMC 12/AHSME, 11

Tags: logarithm

If $\log_2(\log_2(\log_2(x)))=2$, then how many digits are in the base-ten representation for $x$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 13 $

2014 Indonesia MO Shortlist, G3

Tags: geometry , trapezoid , parallelogram , trigonometry , geometric transformation , dilation , logarithm

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

2000 Tuymaada Olympiad, 4

Tags: logarithm , inequalities

Prove for real $x_1$, $x_2$, ....., $x_n$, $0 < x_k \leq {1\over 2}$, the inequality \[ \left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right). \]

2014 PUMaC Combinatorics B, 6

Tags: logarithm

Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$.

1988 Irish Math Olympiad, 4

Tags: trigonometry , geometry , logarithm

Problem: A mathematical moron is given the values b; c; A for a triangle ABC and is required to find a. He does this by using the cosine rule $ a^2 = b^2 + c^2 - 2bccosA$ and misapplying the low of the logarithm to this to get $ log a^2 = log b^2 + log c^2 - log(2bc cos A) $ He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms and gets the correct answer. What can be said about the triangle ABC?

2019 Korea USCM, 5

Tags: series , exponential , logarithm , sequence

A sequence $\{a_n\}_{n\geq 1}$ is defined by a recurrence relation $$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}$$ And a sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate an infinite series $\sum\limits_{n=1}^\infty b_n$.

2005 China Team Selection Test, 2

Tags: algebra , logarithm

Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

2010 Today's Calculation Of Integral, 660

Tags: calculus , integration , calculus computations , logarithm

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

1984 IMO Shortlist, 20

Tags: inequalities , algebra , logarithm

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

2011 AMC 12/AHSME, 19

Tags: floor function , function , inequalities , algebra , logarithm

At a competition with $N$ players, the number of players given elite status is equal to \[2^{1+\lfloor\log_2{(N-1)}\rfloor} - N. \] Suppose that $19$ players are given elite status. What is the sum of the two smallest possible values of $N$? $ \textbf{(A)}\ 38\qquad \textbf{(B)}\ 90 \qquad \textbf{(C)}\ 154 \qquad \textbf{(D)}\ 406 \qquad \textbf{(E)}\ 1024$

2009 Indonesia TST, 2

Tags: parameterization , inequalities , algebra , logarithm

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

1982 Tournament Of Towns, (027) 1

Tags: floor function , radical , logarithm , sum , algebra

Prove that for all natural numbers $n$ greater than $1$ : $$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$$ (VV Kisil)

2013 Moldova Team Selection Test, 4

Tags: limit , algebra , logarithm

Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as: $x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$. $a)$ Prove that the sequence is converging. $b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.

2002 AMC 12/AHSME, 21

Tags: logarithm

Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c$, different from $1$, such that \[2(\log_ac+\log_bc)=9\log_{ab}c.\] Find the largest possible value of $\log_ab$. $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\sqrt3\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }3$

2011 ELMO Shortlist, 5

Tags: algorithm , combinatorics , logarithm

Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where a) $f(n)=n\ln{n}$; b) $f(n)=n$. [i]David Yang.[/i]

2011 Today's Calculation Of Integral, 730

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

Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$. Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

2011 Today's Calculation Of Integral, 737

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

Let $a,\ b$ real numbers such that $a>1,\ b>1.$ Prove the following inequality. \[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]

2012 Today's Calculation Of Integral, 795

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

Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$

1984 AIME Problems, 5

Tags: symmetry , logarithm

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

2012 Balkan MO, 3

Tags: induction , inequalities , floor function , combinatorics , logarithm

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

2003 District Olympiad, 4

1993 AMC 12/AHSME, 11

2014 Indonesia MO Shortlist, G3

2000 Tuymaada Olympiad, 4

2014 PUMaC Combinatorics B, 6

1988 Irish Math Olympiad, 4

2019 Korea USCM, 5

2005 China Team Selection Test, 2

2010 Today's Calculation Of Integral, 660

1984 IMO Shortlist, 20

2011 AMC 12/AHSME, 19

2009 Indonesia TST, 2

1982 Tournament Of Towns, (027) 1

2013 Moldova Team Selection Test, 4

2002 AMC 12/AHSME, 21

2011 ELMO Shortlist, 5

2011 Today's Calculation Of Integral, 730

2011 Today's Calculation Of Integral, 737

2012 Today's Calculation Of Integral, 795

1984 AIME Problems, 5

2012 Balkan MO, 3

PEN A Problems, 107

2011 Today's Calculation Of Integral, 751

2008 Vietnam National Olympiad, 1

2005 Today's Calculation Of Integral, 11