Olympiad problems

2004 Nicolae Coculescu, 2

Tags: quadratics , monotony , equations , algebra , trigonometry , logarithms

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

1970 Czech and Slovak Olympiad III A, 6

Tags: inequalities , trigonometry , algebra , logarithms

Determine all real $x$ such that \[\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.\]

1966 IMO Shortlist, 30

Tags: inequalities , logarithms , calculus , IMO Shortlist , IMO Longlist

Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

2010 Today's Calculation Of Integral, 537

Tags: calculus , integration , trigonometry , logarithms , calculus computations

Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.

2011 Today's Calculation Of Integral, 757

Tags: calculus , integration , logarithms , calculus computations

Evaluate \[\int_0^1 \frac{(x^2+x+1)^3\{\ln (x^2+x+1)+2\}}{(x^2+x+1)^3}(2x+1)e^{x^2+x+1}dx.\]

2010 All-Russian Olympiad, 3

Tags: logarithms , number theory , prime numbers , number theory proposed

Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $.

2010 Malaysia National Olympiad, 2

Tags: logarithms , algebra unsolved , algebra

Find $x$ such that \[2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}\]

2011 Bogdan Stan, 1

Tags: inequalities , algebra , logarithms , logarithm

If $ a,b,c $ are all in the interval $ (0,1) $ or all in the interval $ \left( 1,\infty \right), $ then $$ 1\le\sum_{\text{cyc}} \frac{\log_a^7 b\cdot \log_b^3c}{\log_c a +2\log_a b} . $$ [i]Gheorghe Duță[/i]

1959 AMC 12/AHSME, 11

Tags: logarithms , AMC , logarithm , AMC 12

The logarithm of $.0625$ to the base $2$ is: $ \textbf{(A)}\ .025 \qquad\textbf{(B)}\ .25\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ -4\qquad\textbf{(E)}\ -2 $

2022 Bulgarian Spring Math Competition, Problem 11.1

Tags: algebra , logarithms

Solve the equation \[(x+1)\log^2_{3}x+4x\log_{3}x-16=0\]

1971 IMO Longlists, 53

Tags: limit , floor function , logarithms , number theory proposed , number theory

Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.

2003 CentroAmerican, 3

Tags: inequalities , calculus , logarithms , induction , inequalities proposed

Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.

2010 Today's Calculation Of Integral, 660

Tags: calculus , integration , logarithms , calculus computations

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

2011 Tokyo Instutute Of Technology Entrance Examination, 1

Tags: calculus , integration , trigonometry , geometry , limit , logarithms , calculus computations

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

2003 National High School Mathematics League, 10

Tags: logarithms

$a,b,c,d$ are positive integers, and $\log_{a}b=\frac{3}{2},\log_{c}d=\frac{5}{4}$. If $a-c=9$, then $b-d=$________.

2011 AIME Problems, 9

Tags: trigonometry , logarithms , AMC , AIME , algebra , polynomial , Rational Root Theorem

Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$. Find $24\cot^2{x}$.

2009 China Team Selection Test, 1

Tags: ceiling function , ratio , function , floor function , logarithms , combinatorics proposed , combinatorics

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

1983 IMO Longlists, 32

Tags: function , inequalities , logarithms , algebra unsolved , algebra

Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and \[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\] Prove that if $b + c < 1$, there is a real number $k$ such that \[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\] while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$

2012 China Team Selection Test, 1

Tags: logarithms , floor function , function , number theory proposed , number theory

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 \[\min \{|A|,|B|\}\le\log _2n.\]

1956 AMC 12/AHSME, 20

Tags: logarithms

If $ (0.2)^x \equal{} 2$ and $ \log 2 \equal{} 0.3010$, then the value of $ x$ to the nearest tenth is: $ \textbf{(A)}\ \minus{} 10.0 \qquad\textbf{(B)}\ \minus{} 0.5 \qquad\textbf{(C)}\ \minus{} 0.4 \qquad\textbf{(D)}\ \minus{} 0.2 \qquad\textbf{(E)}\ 10.0$

2007 Today's Calculation Of Integral, 217

Tags: calculus , integration , logarithms , function , geometry , trigonometry , calculus computations

Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.

2008 ISI B.Stat Entrance Exam, 6

Tags: limit , logarithms , integration , calculus , function , real analysis , calculus computations

Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$

2003 Pan African, 1

Tags: function , logarithms , induction

Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that: (1) $f(n) < f(n+1)$, all $n \in N_0$; (2) $f(2)=2$; (3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.

2014 NIMO Summer Contest, 6

Tags: ceiling function , logarithms , floor function , probability , expected value , number theory , relatively prime

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

2008 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: inequalities , logarithms , circumcircle , High school olympiad , Bosnia , algebra

If $ a$, $ b$ and $ c$ are positive reals prove inequality: \[ \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.\]