Olympiad problems

2012 Online Math Open Problems, 28

Tags: modular arithmetic , real analysis , function , limit , number theory , logarithm

Find the remainder when \[\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot 2^{14}+1)^k (k-1)^{2^{16}-1}\]is divided by $2^{16}+1$. ([i]Note:[/i] It is well-known that $2^{16}+1=65537$ is prime.) [i]Victor Wang.[/i]

2009 AIME Problems, 2

Tags: algebra , logarithm

Suppose that $ a$, $ b$, and $ c$ are positive real numbers such that $ a^{\log_3 7} \equal{} 27$, $ b^{\log_7 11} \equal{} 49$, and $ c^{\log_{11} 25} \equal{} \sqrt {11}$. Find \[ a^{(\log_3 7)^2} \plus{} b^{(\log_7 11)^2} \plus{} c^{(\log_{11} 25)^2}. \]

1981 National High School Mathematics League, 8

Tags: logarithm

In the logarithm table below, there are two mistakes. Correct them. \begin{tabular}{|c|c|} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $\lg0.021$&$2a+b+c-3$ \\ \hline $\lg0.27$&$6a-3b-2$\\ \hline $\lg1.5$&$3a-b+c$\\ \hline $\lg2.8$&$1-2a+2b-c$\\ \hline $\lg3$&$2a-b$\\ \hline $\lg5$&$a+c$\\ \hline $\lg6$&$1+a-b-c$\\ \hline $\lg7$&$2(a+c)$\\ \hline $\lg8$&$3-3a-3c$\\ \hline $\lg9$&$4a-2b$\\ \hline $\lg14$&$1-a+2b$\\ \hline \end{tabular}

2010 Victor Vâlcovici, 2

Tags: function , inequalities , calculus , logarithm

Let $ f:[2,\infty )\rightarrow\mathbb{R} $ be a differentiable function satisfying $ f(2)=0 $ and $$ \frac{df}{dx}=\frac{2}{x^2+f^4{x}} , $$ for any $ x\in [2,\infty ) . $ Show that there exists $ \lim_{x\to\infty } f(x) $ and is at most $ \ln 3. $ [i]Gabriel Daniilescu[/i]

2012 Today's Calculation Of Integral, 784

Tags: calculus , integration , function , analytic geometry , geometry , limit , logarithm

Define for positive integer $n$, a function $f_n(x)=\frac{\ln x}{x^n}\ (x>0).$ In the coordinate plane, denote by $S_n$ the area of the figure enclosed by $y=f_n(x)\ (x\leq t)$, the $x$-axis and the line $x=t$ and denote by $T_n$ the area of the rectagle with four vertices $(1,\ 0),\ (t,\ 0),\ (t,\ f_n(t))$ and $(1,\ f_n(t))$. (1) Find the local maximum $f_n(x)$. (2) When $t$ moves in the range of $t>1$, find the value of $t$ for which $T_n(t)-S_n(t)$ is maximized. (3) Find $S_1(t)$ and $S_n(t)\ (n\geq 2)$. (4) For each $n\geq 2$, prove that there exists the only $t>1$ such that $T_n(t)=S_n(t)$. Note that you may use $\lim_{x\to\infty} \frac{\ln x}{x}=0.$

1978 Miklós Schweitzer, 3

Tags: number theory , combinatorics , logarithm

Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $ \sum_{i\equal{}1}^n 1/a_i \leq 1$. Let $ y$ denote the number of positive integers smaller that $ x$ not divisible by any of the $ a_i$. Prove that \[ y > \frac{cx}{\log x}\] with a suitable positive constant $ c$ (independent of $ x$ and the numbers $ a_i$). [i]I. Z. Ruzsa[/i]

1962 AMC 12/AHSME, 17

Tags: logarithm

If $ a \equal{} \log_8 225$ and $ b \equal{} \log_2 15,$ then $ a$, in terms of $ b,$ is: $ \textbf{(A)}\ \frac{b}{2} \qquad \textbf{(B)}\ \frac{2b}{3}\qquad \textbf{(C)}\ b \qquad \textbf{(D)}\ \frac{3b}{2} \qquad \textbf{(E)}\ 2b$

2013 ELMO Shortlist, 9

Tags: function , logarithm , inequalities

Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

2012 Waseda University Entrance Examination, 4

Tags: calculus , integration , function , analytic geometry , calculus computations , logarithm

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

2016 District Olympiad, 1

Tags: equation , algebra , logarithm

Solve in the interval $ (2,\infty ) $ the following equation: $$ 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) . $$

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 $z$?

2012 Kyoto University Entry Examination, 1

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2009 Today's Calculation Of Integral, 439

Tags: calculus , integration , inequalities , analytic geometry , calculus computations , logarithm

Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$. Note that you may not directively use double integral here for Japanese high school students who don't study it.

2004 China Team Selection Test, 2

Tags: floor function , number theory , logarithm

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

2004 Nicolae Coculescu, 2

Tags: equation , monotone , quadratic , algebra , trigonometry , logarithm

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]

2007 Princeton University Math Competition, 10

Tags: logarithm

Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.

2001 District Olympiad, 4

Tags: logarithm , algebra

Solve the equation: \[2^{\lg x}+8=(x-8)^{\frac{1}{\lg 2}}\] Note: $\lg x=\log_{10}x$. [i]Daniel Jinga [/i]

2023 India IMO Training Camp, 2

Tags: modular arithmetic , number theory , logarithm

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 $s(2^n) > s(2^{n+1})$.

2007 Putnam, 3

Tags: floor function , calculus , integration , function , induction , logarithm

Let $ x_0 \equal{} 1$ and for $ n\ge0,$ let $ x_{n \plus{} 1} \equal{} 3x_n \plus{} \left\lfloor x_n\sqrt {5}\right\rfloor.$ In particular, $ x_1 \equal{} 5,\ x_2 \equal{} 26,\ x_3 \equal{} 136,\ x_4 \equal{} 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

1990 AMC 12/AHSME, 23

Tags: logarithm

If $x,y>0$, $\log_yx+\log_xy=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$ $ \textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 13\sqrt{3} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36 $

1995 Irish Math Olympiad, 1

Tags: induction , logarithm , inequalities

Prove that for every positive integer $ n$, $ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$

PEN A Problems, 13

Tags: floor function , inequalities , divisibility theory , logarithm

Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.

2002 AIME Problems, 3

Tags: ratio , geometric sequence , logarithm

It is given that $\log_{6}a+\log_{6}b+\log_{6}c=6,$ where $a,$ $b,$ and $c$ are positive integers that form an increasing geometric sequence and $b-a$ is the square of an integer. Find $a+b+c.$

2017 AIME Problems, 7

Tags: genius title , logarithm

Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\log(kx)=2\log(x+2)$ has exactly one real solution.

2013 ELMO Shortlist, 2

Tags: induction , strong induction , combinatorics , logarithm

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]