Olympiad problems

2010 Math Prize For Girls Problems, 17

Tags: function , logarithm

For every $x \ge -\frac{1}{e}\,$, there is a unique number $W(x) \ge -1$ such that \[ W(x) e^{W(x)} = x. \] The function $W$ is called Lambert's $W$ function. Let $y$ be the unique positive number such that \[ \frac{y}{\log_{2} y} = - \frac{3}{5} \, . \] The value of $y$ is of the form $e^{-W(z \ln 2)}$ for some rational number $z$. What is the value of $z$?

1996 Canadian Open Math Challenge, 9

Tags: logarithm

If $\log_{2n} 1994 = \log_n \left(486 \sqrt{2}\right)$, compute $n^6$.

1994 China Team Selection Test, 1

Tags: logarithm , algebra , inequalities , function , domain , linear algebra , matrix

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

2014 Indonesia MO, 3

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

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 $KL$.

IV Soros Olympiad 1997 - 98 (Russia), 11.7

Tags: inequalities , logarithm , algebra

Solve the inequality $$\log_{\frac12} x\ge 16^x$$

2005 Today's Calculation Of Integral, 39

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

Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$. \[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]

2011 ELMO Shortlist, 5

Tags: algorithm , logarithm , combinatorics

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]

2019 AIME Problems, 6

Tags: algebra , system of equations , logarithm

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down \begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54 \end{align*} and finds that this system of equations has a single real number solution $x > 1$. Find $b$.

2012 Balkan MO Shortlist, C1

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

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

2010 Contests, 3

Tags: floor function , combinatorics , logarithm , induction

There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$. For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$. Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible). Sorry for my bad English.

1991 AMC 12/AHSME, 24

Tags: geometric transformation , rotation , logarithm , geometry

The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$? $ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $

2008 Regional Competition For Advanced Students, 4

Tags: number theory , logarithm , modular arithmetic , floor function

For every positive integer $ n$ let \[ a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k}\] Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.

2011 Today's Calculation Of Integral, 711

Tags: calculus , integration , calculus computations , logarithm

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

2019 Jozsef Wildt International Math Competition, W. 6

Tags: trigonometry , integration , calculus , logarithm

Compute$$\int \limits_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{(1+\ln x)\cos x+x\sin x\ln x}{\cos^2 x + x^2 \ln^2 x}dx$$

1994 APMO, 5

Tags: floor function , number theory , logarithm

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.

2011 District Round (Round II), 1

Tags: number theory , logarithm

Among all eight-digit multiples of four, are there more numbers with the digit $1$ or without the digit $1$ in their decimal representation?

2004 Romania National Olympiad, 2

Tags: function , polynomial , inequalities , calculus , algebra , logarithm , integration

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$. [i]Laurentiu Panaitopol[/i]

1950 AMC 12/AHSME, 44

Tags: logarithm

The graph of $ y\equal{}\log x$ $\textbf{(A)}\text{Cuts the }y\text{-axis} \qquad\\ \textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\ \textbf{(C)}\ \text{Cuts the }x\text{-axis} \qquad\\ \textbf{(D)}\ \text{Cuts neither axis} \qquad\\ \textbf{(E)}\ \text{Cuts all circles whose center is at the origin}$

1978 IMO Longlists, 37

Tags: logarithm , algebra

Simplify \[\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},\] where $a, b, c$ are positive real numbers.

2009 Iran Team Selection Test, 12

Tags: logarithm , combinatorics

$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that : $ |T| \leq \frac {4}{9}n + \log_2 n + 2$

1956 AMC 12/AHSME, 20

Tags: logarithm

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$

2012 IMC, 3

Tags: logarithm

Is the set of positive integers $n$ such that $n!+1$ divides $(2012n)!$ finite or infinite? [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

1969 AMC 12/AHSME, 29

Tags: logarithm

If $x=t^{(1/(t-1))}$ and $x=t^{(t/(t-1))}$, $t>0$, $t\not=1$, a relation between $x$ and $y$ is $\textbf{(A)}\ y^x=x^{1/y}\qquad \textbf{(B)}\ y^{1/x}=x^{y} \qquad \textbf{(C)}\ y^x=x^{y}\qquad \textbf{(D)}\ x^x=y^y\\ \textbf{(E)}\ \text{none of these}$

2006 China Second Round Olympiad, 5

Tags: logarithm

Suppose $f(x) = x^3 + \log_2(x + \sqrt{x^2+1})$. For any $a,b \in \mathbb{R}$, to satisfy $f(a) + f(b) \ge 0$, the condition $a + b \ge 0$ is $ \textbf{(A)}\ \text{necessary and sufficient}\qquad\textbf{(B)}\ \text{not necessary but sufficient}\qquad\textbf{(C)}\ \text{necessary but not sufficient}\qquad$ $\textbf{(D)}\ \text{neither necessary nor sufficient}\qquad$

2012 India Regional Mathematical Olympiad, 3

Tags: inequalities , function , logarithm

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.