Found problems: 913
Problem: A mathematical moron is given the values b; c; A for a triangle ABC and
is required to fi nd 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?
Suppose that $ 4^{x_1} \equal{} 5, 5^{x_2} \equal{} 6, 6^{x_3} \equal{} 7,...,127^{x_{124}} \equal{} 128$. What is $ x_1x_2 \cdots x_{124}$?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ \frac {5}{2}\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ \frac {7}{2}\qquad
\textbf{(E)}\ 4$
Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$.
Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions.
(1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value.
(2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.
Solve in real numbers the equation
\[ x+\log_{2}\left( 1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right) =4+\log_{1/2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right) \]
Answer either (i) or (ii):
(i) Prove that
$$\sum_{n=2}^{\infty} \frac{\cos (\log \log n)}{\log n}$$
diverges.
(ii) Assume that $p>0, a>0$, and $ac-b^{2} >0,$ and show that
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{ dx\; dy}{(p+ax^2 +2bxy+ cy^2 )^{2}}= \pi p^{-1} (ac-b^{2})^{- 1\slash 2}.$$
Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.
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]
Prove that : $\frac{1}{(\log_{bc} a)^n}+\frac{1}{(\log_{ac} b)^n}+\frac{1}{(\log_{bc} a)^n}\geq 3\cdot2^{n}$ where $a,b,c>1$ and $n$ is natural number.
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$
Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$
Let $ (\log_2(x))^2 \minus{} 4 \cdot \log_2(x) \minus{} m^2 \minus{} 2m \minus{} 13 \equal{} 0$ be an equation in $ x.$ Prove:
[b](a)[/b] For any real value of $ m$ the equation has two distinct solutions.
[b](b)[/b] The product of the solutions of the equation does not depend on $ m.$
[b](c)[/b] One of the solutions of the equation is less than 1, while the other solution is greater than 1.
Find the minimum value of the larger solution and the maximum value of the smaller solution.
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]
For a positive integer $n$, let
\[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\]
Answer the following questions:
(1) Show the following inequality.
\[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\]
(2) Express $T_n-2S_n$ in terms of $n$.
(3) Find the limit $\lim_{n\to\infty} T_n.$
Find $\lim_{n\to\infty} \frac{1}{n}\sqrt[n]{\frac{(4n)!}{(3n)!}}.$
If $\log_{2n} 1994 = \log_n \left(486 \sqrt{2}\right)$, compute $n^6$.
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\sin ^ 2 x}{\cos ^ 3 x}\ dx$.
Compute the sum of the series
$\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$
$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.
Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis.
[i]1993 Hiroshima Un iversity entrance exam/Science[/i]
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]
The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$.
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$