Found problems: 85335
With three different digits, six-digit numbers are written, multiples of $3$. One of the the digits are in the unit's place, another in the hundred's place, and the third in the remaining places. If we take out two units from the hundred's digit and add these to the unit's digit, the number is left with all the same digits. Find the numbers.
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect in point $M$. The angle bisector of $\angle ACD$ intersects the ray $\overrightarrow{BA}$ in point $K$. If
$MA.MC+MA.CD=MB.MD$,
prove that $\angle BKC=\angle CDB$.
A computer program generated $175$ positive integers at random, none of which had a prime divisor grater than $10.$ Prove that there are three numbers among them whose product is the cube of an integer.
Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?
Compute the integral
\[\iint_{\mathbb{R}^2} \left(\frac{1-e^{-xy}}{xy}\right)^2 e^{-x^2-y^2} dx dy.\]
For a given integer $n\geq2$, a pyramid of height $n$ if defined as a collection of $1^2+2^2+\dots+n^2$ stone cubes of equal size stacked in $n$ layers such that the cubes in the $k$-th layer form a square with sidelength $n+1-k$ and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?
The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure?
[img]https://1.bp.blogspot.com/-oWnh8uhyTIo/XzP30gZueKI/AAAAAAAAMUY/GlC3puNU_6g6YRf6hPpbQW8IE8IqMP3ugCLcBGAsYHQ/s0/2018%2BMohr%2Bp2.png[/img]
Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?
Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.)
[i]Boris Frenkin[/i]
For all positive real numbers $a,b,c$, prove that
$$\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)$$
Let $ f: \mathbb{R}\rightarrow \mathbb{R}$ a strictly increasing function such that $ f\circ f$ is continuos. Prove that $ f$ is continuos.
In a $3$ by $3$ table, by a $k$-worm, we mean a path of different cells $(S_1,S_2,...,S_k)$ such that each two consecutive cells have one side in common. The $k$-worm at each steep can go one cell forward and turn to the $(S,S_1,...,S_{k-1})$ if $S$ is an unfilled cell which is adjacent (has one side in common) with $S_1$. Find the maximum number of $k$ such that there is a $k$-worm $(S_1,...,S_k)$ such that after finitly many steps can be turned to $(S_k,...,S_1)$.
The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.
Let $f(x)$ be a polynomial such that $2f(x) + f(2 - x) = 5 + x$ for any real number x. Find the value of $f(0) + f(2)$.
A. $4$ B. $0$ C.$ 2$ D. $3$ E. $1$
A straight cone is given inside a rectangular parallelepiped $B$, with the apex at one of the vertices, say $T$ , of the parallelepiped, and the base touching the three faces opposite to $T .$ Its axis lies at the long diagonal through $T.$ If $V_1$ and $V_2$ are the volumes of the cone and the parallelepiped respectively, prove that
\[V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.\]
Equilateral triangle $A_1A_2A_3$ has side length $15$ and circumcenter $M$. Let $N$ be a point such that $\angle A_3MN = 72^{\circ}$ and $MN = 7$. The circle with diameter $MN$ intersects lines $MA_1$, $MA_2$, and $MA_3$ again at $B_1$, $B_2$, and $B_3$, respectively. The value of $NB_1^2+NB_2^2+NB_3^2$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Tiebreaker #4[/i]
A circle is centered at point $O$ in the plane. Distinct pairs of points $A, B$ and $C, D$ are diametrically opposite on this circle. Point $P$ is chosen on line segment $AD$ such that line $BP$ hits the circle again at $M$ and line $AC$ at $X$ such that $M$ is the midpoint of $PX$. Now, the point $Y \neq X$ is taken for $BX = BY, CD \parallel XY$. IF $\angle PYB = 10^{\circ}$, find the measure of $\angle XCM$.
Proposed by Albert Wang (awang11)
Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?
[i]Proposed by Matthew Weiss
Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that:
[i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$
[i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$
[i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$
Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.
Let $x_1,x_2,\cdots,x_n$ be postive real numbers such that $x_1x_2\cdots x_n=1$ ,$S=x^3_1+x^3_2+\cdots+x^3_n$.Find the maximum of $\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}$
A $50$-card deck consists of $4$ cards labeled " i" for $i = 1, 2,..., 12$ and $2$ cards labeled "$13$". If Bob randomly chooses $2$ cards from the deck without replacement, what is the probability that his $2$ cards have the same label?
Let $K$ be a convex shape in the $n$ dimensional space, having unit volume. Let $S \subset K$ be a Lebesgue measurable set with measure at least $1-\varepsilon$, where $0<\varepsilon<1/3$. Prove that dilating $K$ from its centroid by the ratio of $2\varepsilon \ln(1/\varepsilon)$, the shape obtained contains the centroid of $S$.
Let $M$ be a finite set of real numbers with the following property: From three different elements of $M$ can always be chosen two whose sum is located in $M$. How many elements can $M$ have at most?
The numbers $1,2,...,100$ are written in a board. We are allowed to choose any two numbers from the board $a,b$ to delete them and replace on the board the number $a+b-1$.
What are the possible numbers u can get after $99$ consecutive operations of these?