Olympiad problems

2011 Tokyo Instutute Of Technology Entrance Examination, 1

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

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 $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

1957 Putnam, A6

Tags: limit , logarithm

Let $a>0$, $S_1 =\ln a$ and $S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )$ for $n >1.$ Show that $$ \lim_{n \to \infty} S_n = a-1.$$

1954 AMC 12/AHSME, 38

Tags: logarithm

If $ \log 2\equal{}.3010$ and $ \log 3\equal{}.4771$, the value of $ x$ when $ 3^{x\plus{}3}\equal{}135$ is approximately: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 1.47 \qquad \textbf{(C)}\ 1.67 \qquad \textbf{(D)}\ 1.78 \qquad \textbf{(E)}\ 1.63$

1972 Swedish Mathematical Competition, 4

Tags: algebra , compare , inequalities , logarithm

Put $x = \log_{10} 2$, $y = \log_{10} 3$. Then $15 < 16$ implies $1 - x + y < 4x$, so $1 + y < 5x$. Derive similar inequalities from $80 < 81$ and $243 < 250$. Hence show that \[ 0.47 < \log_{10} 3 < 0.482. \]

1998 Greece JBMO TST, 4

Tags: polynomial , algebra , logarithm

(a) A polynomial $P(x)$ with integer coefficients takes the value $-2$ for at least seven distinct integers $x$. Prove that it cannot take the value $1996$. (b) Prove that there are irrational numbers $x,y$ such that $x^y$ is rational.

Today's calculation of integrals, 896

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

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

2019 Korea USCM, 5

Tags: series , exponential , logarithm , sequence

A sequence $\{a_n\}_{n\geq 1}$ is defined by a recurrence relation $$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}$$ And a sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate an infinite series $\sum\limits_{n=1}^\infty b_n$.

1950 Miklós Schweitzer, 7

Tags: integration , calculus , euler , function , logarithm , real analysis

Examine the behavior of the expression $ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$ as $ n\rightarrow \infty$

2005 Today's Calculation Of Integral, 23

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

Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

1961 AMC 12/AHSME, 19

Tags: function , algebra , domain , logarithm

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that: $ \textbf{(A)}\ \text{They do not intersect}$ $ \qquad\textbf{(B)}\ \text{They intersect at 1 point only}$ $\qquad\textbf{(C)}\ \text{They intersect at 2 points only}$ $\qquad\textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }$ ${\qquad\textbf{(E)}\ \text{They coincide} } $

1998 Estonia National Olympiad, 1

Tags: algebra , logarithm

Solve the equation $x^2+1 = log_2(x+2)- 2x$.

1958 AMC 12/AHSME, 25

Tags: logarithm

If $ \log_{k}{x}\cdot \log_{5}{k} \equal{} 3$, then $ x$ equals: $ \textbf{(A)}\ k^6\qquad \textbf{(B)}\ 5k^3\qquad \textbf{(C)}\ k^3\qquad \textbf{(D)}\ 243\qquad \textbf{(E)}\ 125$

2013 Today's Calculation Of Integral, 893

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

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

PEN E Problems, 24

Tags: floor function , limit , number theory , prime number , logarithm , inequalities

Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.

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?

1997 National High School Mathematics League, 12

Tags: logarithm

Let $a=\lg z+\lg\left[x(yz)^{-1}+1\right],b=\lg x^{-1}+\lg(xyz+1),c=\lg y+\lg\left[(xyz)^{-1}+1\right]$, if $M=\max\{a,b,c\}$, then the minumum value of $M$ is________.

2010 Today's Calculation Of Integral, 560

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

Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$. (1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis. (2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis. Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.

1976 AMC 12/AHSME, 20

Tags: quadratic , logarithm

Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\] $\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$ $\textbf{(B) }\text{if and only if }a=b^2\qquad$ $\textbf{(C) }\text{if and only if }b=a^2\qquad$ $\textbf{(D) }\text{if and only if }x=ab\qquad$ $ \textbf{(E) }\text{for none of these}$

1984 IMO Shortlist, 20

Tags: inequalities , algebra , logarithm

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

2010 Polish MO Finals, 3

Tags: limit , algebra , inequality , inequalities , logarithm

Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions \[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\] for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.

2011 Today's Calculation Of Integral, 761

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

Find $\lim_{n\to\infty} \frac{1}{n}\sqrt[n]{\frac{(4n)!}{(3n)!}}.$

1993 AMC 12/AHSME, 11

Tags: logarithm

If $\log_2(\log_2(\log_2(x)))=2$, then how many digits are in the base-ten representation for $x$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 13 $

2005 China National Olympiad, 4

Tags: inequalities , calculus , function , derivative , logarithm , algebra

The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]

2012 Romania Team Selection Test, 1

Tags: floor function , function , induction , algebra , logarithm

Prove that for any positive integer $n\geq 2$ we have that \[\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.\]

1953 AMC 12/AHSME, 39

Tags: logarithm

The product, $ \log_a b \cdot \log_b a$ is equal to: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ a \qquad\textbf{(C)}\ b \qquad\textbf{(D)}\ ab \qquad\textbf{(E)}\ \text{none of these}$