Found problems: 85335
Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real $x, y$ holds equality
$$f(xf(y)) + f(xy) = 2f(x)y$$
[i]Proposed by Arseniy Nikolaev[/i]
Three metals, $A, B $and $C$, with solutions of their respective cations are tested in a voltaic cell with the following results:
$A$ and $B$: $A$ is the cathode
$B$ and $C$: $C$ is the cathode
$A$ and $C$: $A$ is the anode
What is the order of the reduction potentials from highest to lowest for the cations of these metals?
$ \textbf{(A)}\ A>B>C \qquad\textbf{(B)}\ B>C>A\qquad$
${\textbf{(C)}\ C>A>B\qquad\textbf{(D}}\ B>A>C\qquad$
$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space.
1) Prove that Image of every line is a line.
2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)
An integer $N$ is selected at random in the range $1\le N \le 2020.$ What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
$\textbf{(A)} \text{ }\frac{1}{5} \qquad \textbf{(B)} \text{ }\frac{2}{5} \qquad \textbf{(C)} \text{ }\frac{3}{5} \qquad \textbf{(D)} \text{ }\frac{4}{5} \qquad \textbf{(E)} \text{ 1}$
Two different points $A, M$ are given in a plane, $AM = d > 0$. Let a number $v > 0$ be given. Construct a rhombus $ABCD$ with the height of length $v$ and $M$ being a midpoint of $BC$. Discuss conditions of solvability and determine number of solutions. Can the resulting quadrilateral $ABCD$ be a square?
What is the value of $$\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$$
$\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000$
There is a point set on a plane, and seven circles $C_1,C_2,\cdots,C_7$, where $C_7$ passes exactly 7 points in $M$, $C_6$ passes exactly 6 points in $M$, ..., $C_1$ passes exactly 1 point in $M$. Then how many points do set $M$ have at least?
$\text{(A)}11\qquad\text{(B)}12\qquad\text{(C)}21\qquad\text{(D)}28$
A sequence of real numbers $\{a_n\}_n$ is called a [i]bs[/i] sequence if $a_n = |a_{n+1} - a_{n+2}|$, for all $n\geq 0$. Prove that a bs sequence is bounded if and only if the function $f$ given by $f(n,k)=a_na_k(a_n-a_k)$, for all $n,k\geq 0$ is the null function.
[i]Mihai Baluna - ISL 2004[/i]
Let
$ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients.
Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.
Convex hexagon $A_1A_2A_3A_4A_5A_6$ lies in the interior of convex hexagon $B_1B_2B_3B_4B_5B_6$ such that $A_1A_2 \parallel B_1B_2$, $A_2A_3 \parallel B_2B_3$,..., $A_6A_1 \parallel B_6B_1$. Prove that the areas of simple hexagons $A_1B_2A_3B_4A_5B_6$ and $B_1A_2B_3A_4B_5A_6$ are equal. (A simple hexagon is a hexagon which does not intersect itself.)
[i]Proposed by Hirad Aalipanah - Mahdi Etesamifard[/i]
In a convex quadrilateral $ABCD, \angle A < 90^o, \angle B < 90^o$ and $AB > CD$. Points $P$ and $Q$ are on the segments $BC$ and $AD$ respectively. Suppose the triangles $APD$ and $BQC$ are similar. Prove that $AB$ is parallel to $CD$.
Points $A,B,C$ with $AB = BC$ are given on a circle with radius $r$, and $D$ is a point inside the circle such that the triangle $BCD$ is equilateral. The line $AD$ meets the circle again at $E$. Show that $DE = r$.
Prove that if two medians in a triangle are equal in length, then the triangle is isosceles.
(Note: A median in a triangle is a segment which connects a vertex of the triangle to the midpoint of the opposite side of the triangle.)
The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then
the number of possible coverings is even.
[i]D. Karpov, C. Gukshin, D. Fon-der-Flaas[/i]
Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?
If \[\frac{1}{1+2} + \frac{1}{1+2+3} + \ldots + \frac{1}{1+2 + \ldots + 20} = \frac{m}{n}\]
where $m$ and $n$ are positive integers with no common divisor, find $m + n$.
Find all the (x,y) integer ,if
$y^2+2y=x^4+20x^3+104x^2+40x+2003$
Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$
What is the sum of all two-digit positive integer $n<50$ for which the sum of the squares of first $n$ positive integers is not a divisor of $(2n)!$ ?
What is the least integer $n\geq 100$ such that $77$ divides $1+2+2^2+2^3+\dots + 2^n$ ?
$ \textbf{(A)}\ 101
\qquad\textbf{(B)}\ 105
\qquad\textbf{(C)}\ 111
\qquad\textbf{(D)}\ 119
\qquad\textbf{(E)}\ \text{None}
$
A triangle with area $60\text{ units}^2$ has vertices with coordinates of $(-15,x)$, $(0,x)$, and $(25,0)$. Find the largest possible value of $x$.
$\textbf{(A) } {-}8\qquad\textbf{(B) } {-}4\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 16$
Show that no non-zero integers $a$, $b$, $x$, $y$ satisfy
$$
\begin{cases}
a x - b y = 16,\\
a y + b x = 1.
\end{cases}
$$
Let $a$ and $b$ be positive real numbers, with $a < b$ and let $n$ be a positive integer. Prove that for all real numbers $x_1, x_2, \ldots , x_n \in [a, b]$:
$$ |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n)$$
And determine for what values of $n$ and $x_1, x_2, \ldots , x_n$ the equality holds.
We have an infinite sequence of integers $\{x_n\}$, such that $x_1 = 1$, and, for all $n \ge 1$, it holds that $x_n < x_{n+1} \le 2n$. Prove that there are two terms of the sequence,$ x_r$ and $x_s$, such that $x_r - x_s = 2018$.