Olympiad problems

2018 District Olympiad, 1

Tags: logarithm

Find $x\in\mathbb{R}$ for which \[\log_2(x^2 + 4) - \log_2x + x^2 - 4x + 2 = 0.\]

2014 Singapore Senior Math Olympiad, 11

Tags: logarithm

Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$. Find the value of $2^{-(1+\log_23)x}$

PEN M Problems, 15

Tags: modular arithmetic , recursive sequences , logarithm

For a given positive integer $k$ denote the square of the sum of its digits by $f_{1}(k)$ and let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of $f_{1991}(2^{1990})$.

1987 India National Olympiad, 1

Tags: relatively prime , algebra , logarithm , number theory

Given $ m$ and $ n$ as relatively prime positive integers greater than one, show that \[ \frac{\log_{10} m}{\log_{10} n}\] is not a rational number.

1994 APMO, 5

Tags: floor function , logarithm , number theory

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.

1954 AMC 12/AHSME, 15

Tags: logarithm

$ \log 125$ equals: $ \textbf{(A)}\ 100 \log 1.25 \qquad \textbf{(B)}\ 5 \log 3 \qquad \textbf{(C)}\ 3 \log 25$ $ \textbf{(D)}\ 3 \minus{} 3\log 2 \qquad \textbf{(E)}\ (\log 25)(\log 5)$

2009 Indonesia TST, 2

Tags: parameterization , inequalities , logarithm , algebra

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

1961 AMC 12/AHSME, 19

Tags: function , algebra , domain , logarithm

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that: $ \textbf{(A)}\ \text{They do not intersect}$ $ \qquad\textbf{(B)}\ \text{They intersect at 1 point only}$ $\qquad\textbf{(C)}\ \text{They intersect at 2 points only}$ $\qquad\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }$ ${\qquad\textbf{(E)}\ \text{They coincide} } $

2011 Today's Calculation Of Integral, 734

Tags: calculus , integration , calculus computations , logarithm

Find the extremum of $f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)$.

2003 AMC 12-AHSME, 17

Tags: function , logarithm

If $ \log(xy^3)\equal{}1$ and $ \log(x^2y)\equal{}1$, what is $ \log(xy)$? $ \textbf{(A)}\ \minus{}\!\frac{1}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{3}{5} \qquad \textbf{(E)}\ 1$

1979 IMO Longlists, 9

Tags: inequalities , function , logarithm

The real numbers $\alpha_1 , \alpha_2, \alpha_3, \ldots, \alpha_n$ are positive. Let us denote by $h = \frac{n}{1/\alpha_1 + 1/\alpha_2 + \cdots + 1/\alpha_n}$ the harmonic mean, $g=\sqrt[n]{\alpha_1\alpha_2\cdots \alpha_n}$ the geometric mean, and $a=\frac{\alpha_1+\alpha_2+\cdots + \alpha_n}{n}$ the arithmetic mean. Prove that $h \leq g \leq a$, and that each of the equalities implies the other one.

V Soros Olympiad 1998 - 99 (Russia), 11.1

Tags: algebra , trigonometry , logarithm

Find at least one root of the equation$$\sin(2 \log_2 x) + tg(3\log_2 x) = \sin6+tg9$$less than $0.01$.

2011 Pre-Preparation Course Examination, 3

Tags: logarithm , advanced fields

prove that $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\pi}{4}$

2014 NIMO Problems, 4

Tags: modular arithmetic , euler , floor function , logarithm

Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.

2011 International Zhautykov Olympiad, 3

Tags: modular arithmetic , induction , number theory , logarithm

Let $\mathbb{N}$ denote the set of all positive integers. An ordered pair $(a;b)$ of numbers $a,b\in\mathbb{N}$ is called [i]interesting[/i], if for any $n\in\mathbb{N}$ there exists $k\in\mathbb{N}$ such that the number $a^k+b$ is divisible by $2^n$. Find all [i]interesting[/i] ordered pairs of numbers.

2005 Today's Calculation Of Integral, 42

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

Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

1994 Turkey Team Selection Test, 2

Tags: limit , algebra , logarithm

Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$

2023 India IMO Training Camp, 2

Tags: modular arithmetic , logarithm , number theory

For a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Show that there are infinitely many natural numbers $n$ such that $s(2^n) > s(2^{n+1})$.

2015 China National Olympiad, 1

Tags: logarithm , number theory

Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying \[n+k |\binom{2n}{n}\]

2012 JBMO TST - Turkey, 2

Tags: induction , floor function , combinatorics , logarithm

Let $S=\{1,2,3,\ldots,2012\}.$ We want to partition $S$ into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of $2.$ Find the number of such partitions.

2022 District Olympiad, P1

Tags: algebra , logarithm

Determine all $x\in(0,3/4)$ which satisfy \[\log_x(1-x)+\log_2\frac{1-x}{x}=\frac{1}{(\log_2x)^2}.\]

2010 District Olympiad, 1

Tags: floor function , algebra , logarithm

Prove the following equalities of sets: \[ \text{i)} \{x\in \mathbb{R}\ |\ \log_2 \lfloor x \rfloor \equal{} \lfloor \log_2 x\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[2^m,2^m \plus{} 1\right)\] \[ \text{ii)} \{x\in \mathbb{R}\ |\ 2^{\lfloor x\rfloor} \equal{} \left\lfloor 2^x\right\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[m, \log_2 (2^m \plus{} 1) \right)\]

1984 AMC 12/AHSME, 14

Tags: logarithm

The product of all real roots of the equation $x^{\log_{10} x} = 10$ is A. 1 B. -1 C. 10 D. $10^{-1}$ E. None of these

1999 Brazil Team Selection Test, Problem 4

Tags: function , inequalities , induction , ring theory , triangle inequality , logarithm , number theory

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

2011 Pre-Preparation Course Examination, 6

Tags: probability , combinatorics , logarithm

We call a subset $S$ of vertices of graph $G$, $2$-dominating, if and only if for every vertex $v\notin S,v\in G$, $v$ has at least two neighbors in $S$. prove that every $r$-regular $(r\ge3)$ graph has a $2$-dominating set with size at most $\frac{n(1+\ln(r))}{r}$.(15 points) time of this exam was 3 hours