Olympiad problems

2010 Today's Calculation Of Integral, 555

Tags: calculus , integration , logarithms , derivative , function , calculus computations

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

2010 Today's Calculation Of Integral, 578

Tags: calculus , integration , inequalities , logarithms , function , algebra , domain

Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$. \[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\] If necessary, you may use $ \ln 3 \equal{} 1.10$.

1994 Vietnam Team Selection Test, 2

Tags: function , logarithms , algebra unsolved , algebra

Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying \[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\] for all $x$.

1995 Canada National Olympiad, 2

Tags: inequalities , logarithms , inequalities proposed

Let $\{a,b,c\}\in \mathbb{R}^{+}$. Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$.

2007 Tuymaada Olympiad, 1

Tags: ratio , logarithms , modular arithmetic , combinatorics proposed , combinatorics

What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?

Today's calculation of integrals, 885

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

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 $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

2010 Iran Team Selection Test, 1

Tags: function , logarithms , number theory , prime numbers , arithmetic sequence , number theory proposed

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

2013 Miklós Schweitzer, 12

Tags: inequalities , ratio , logarithms , probability and stats , Miklos Schweitzer

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let \[ \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.\] Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$ [i]Proposed by Tamás Móri[/i]

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.$

2024 AMC 12/AHSME, 8

Tags: AMC , AMC 12 , 2024 AMC 12A , 2024 AMC , logarithms , trigonometry

How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$? $ \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad $

2012 Balkan MO, 3

Tags: induction , inequalities , floor function , logarithms , combinatorics unsolved , combinatorics

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 $0 \leq y-S_Y < 2^n$

2011 Today's Calculation Of Integral, 743

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

Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$

2009 Today's Calculation Of Integral, 446

Tags: calculus , integration , logarithms , calculus computations

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

2013 Korea - Final Round, 3

Tags: floor function , logarithms , inequalities proposed , inequalities

For a positive integer $n \ge 2 $, define set $ T = \{ (i,j) | 1 \le i < j \le n , i | j \} $. For nonnegative real numbers $ x_1 , x_2 , \cdots , x_n $ with $ x_1 + x_2 + \cdots + x_n = 1 $, find the maximum value of \[ \sum_{(i,j) \in T} x_i x_j \] in terms of $n$.

1969 Miklós Schweitzer, 4

Tags: inequalities , logarithms , real analysis , real analysis unsolved

Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$ \[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} . \] [i]L. Losonczi[/i]

2002 AMC 12/AHSME, 21

Tags: logarithms

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 $\log_ab$. $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\sqrt3\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }3$

2005 Today's Calculation Of Integral, 33

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

Evaluate \[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]

1999 Federal Competition For Advanced Students, Part 2, 1

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

Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.

1992 IMO Longlists, 74

Tags: logarithms , pigeonhole principle , algebra , calculus , limit , IMO Shortlist , IMO Longlist

Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$

2011 Today's Calculation Of Integral, 735

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

Evaluate the following definite integrals: (a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$ (b) $\int_0^{\frac 13} xe^{3x}\ dx$ (c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$ (d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$

2011 Today's Calculation Of Integral, 712