Olympiad problems

2012 Pre - Vietnam Mathematical Olympiad, 1

For $a,b,c>0: \; abc=1$ prove that \[a^3+b^3+c^3+6 \ge (a+b+c)^2\]

2006 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , integration , logarithm

Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.

1996 AMC 12/AHSME, 8

Tags: logarithm

If $3 = k \cdot 2^r$ and $15 = k \cdot 4^r$, then $r =$ $\text{(A)}\ - \log_2 5 \qquad \text{(B)}\ \log_5 2 \qquad \text{(C)}\ \log_{10} 5 \qquad \text{(D)}\ \log_2 5 \qquad \text{(E)}\ \displaystyle \frac{5}{2}$

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?

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.

1988 Iran MO (2nd round), 3

Tags: number theory , logarithm

Let $n$ be a positive integer. $1369^n$ positive rational numbers are given with this property: if we remove one of the numbers, then we can divide remain numbers into $1368$ sets with equal number of elements such that the product of the numbers of the sets be equal. Prove that all of the numbers are equal.

2010 Today's Calculation Of Integral, 615

Tags: calculus , integration , derivative , calculus computations , logarithm

For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ [i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]

2012 ELMO Shortlist, 2

Tags: floor function , ceiling function , algorithm , combinatorics , logarithm

Determine whether it's possible to cover a $K_{2012}$ with a) 1000 $K_{1006}$'s; b) 1000 $K_{1006,1006}$'s. [i]David Yang.[/i]

2012 IberoAmerican, 3

Tags: floor function , number theory , logarithm

Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits. (Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)

2013 Today's Calculation Of Integral, 898

Tags: calculus , integration , calculus computations , logarithm

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

2009 Today's Calculation Of Integral, 461

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

Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$. (1) Find $ I_1,\ I_2$. (2) Find $ \lim_{n\to\infty} I_n$.

1965 AMC 12/AHSME, 21

Tags: limit , function , logarithm

It is possible to choose $ x > \frac {2}{3}$ in such a way that the value of $ \log_{10}(x^2 \plus{} 3) \minus{} 2 \log_{10}x$ is $ \textbf{(A)}\ \text{negative} \qquad \textbf{(B)}\ \text{zero} \qquad \textbf{(C)}\ \text{one}$ $ \textbf{(D)}\ \text{smaller than any positive number that might be specified}$ $ \textbf{(E)}\ \text{greater than any positive number that might be specified}$

2011 Today's Calculation Of Integral, 711

Tags: calculus , integration , calculus computations , logarithm

Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$

1993 Brazil National Olympiad, 5

Tags: function , functional equation , algebra , logarithm

Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$.

2009 Indonesia TST, 2

Tags: inequalities , relatively prime , number theory , logarithm

For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.

2014 Harvard-MIT Mathematics Tournament, 10

Tags: function , calculus , derivative , logarithm

Fix a positive real number $c>1$ and positive integer $n$. Initially, a blackboard contains the numbers $1,c,\ldots, c^{n-1}$. Every minute, Bob chooses two numbers $a,b$ on the board and replaces them with $ca+c^2b$. Prove that after $n-1$ minutes, the blackboard contains a single number no less than \[\left(\dfrac{c^{n/L}-1}{c^{1/L}-1}\right)^L,\] where $\phi=\tfrac{1+\sqrt 5}2$ and $L=1+\log_\phi(c)$.

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 $k$.

2010 Today's Calculation Of Integral, 634

Tags: calculus , integration , calculus computations , logarithm

Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^m\ (n=1,\ 2,\ \cdots)\] [i]2010 Miyazaki University entrance exam/Medicine[/i]

2025 VJIMC, 3

Tags: calculus , integration , logarithm

Evaluate the integral $\int_0^{\infty} \frac{\log(x+2)}{x^2+3x+2}\mathrm{d}x$.

2010 Today's Calculation Of Integral, 576

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

For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions. (1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$. (2) Find the values of $ a,\ b$. (3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.

2006 China Second Round Olympiad, 2

Tags: logarithm

Suppose $log_x (2x^2+x-1)>log_x 2-1$. Then the range of $x$ is ${ \textbf{(A)}\ \frac{1}{2}<x<1\qquad\textbf{(B)}\ x>\frac{1}{2} \text{and} x \not= 1\qquad\textbf{(C)}\ x>1\qquad\textbf{(D)}}\ 0<x<1\qquad $

2007 Ukraine Team Selection Test, 3

Tags: polynomial , algebra , logarithm

It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.

1972 AMC 12/AHSME, 6

Tags: logarithm

If $3^{2x}+9=10(3^{x})$, then the value of $(x^2+1)$ is $\textbf{(A) }1\text{ only}\qquad\textbf{(B) }5\text{ only}\qquad\textbf{(C) }1\text{ or }5\qquad\textbf{(D) }2\qquad \textbf{(E) }10$

1999 National High School Mathematics League, 3

Tags: logarithm

If $(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}$, then $\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0$

2019 PUMaC Team Round, 11

Tags: floor function , combinatorics , logarithm

The game Prongle is played with a special deck of cards: on each card is a nonempty set of distinct colors. No two cards in the deck contain the exact same set of colors. In this game, a “Prongle” is a set of at least $2$ cards such that each color is on an even number of cards in the set. Let k be the maximum possible number of prongles in a set of $2019$ cards. Find $\lfloor \log 2 (k) \rfloor$.