Found problems: 85335
A [i]slip[/i] on an integer $n\geq 2$ is an operation that consists in choosing a prime divisor $p$ of $n$ and replacing $n$ by $\frac{n+p^2}{p}.$
Starting with an arbitrary integer $n\geq 5$, we successively apply the slip operation on it. Show that one eventually reaches $5$, no matter the slips applied.
Let $n$ be a positive integer. Prove that the following two statements are equivalent. [list][*] $n$ is not divisible by $4$ [*] There exist $a, b \in \mathbb{Z}$ such that $a^{2}+b^{2}+1$ is divisible by $n$. [/list]
Let $n> 2$ be an integer. A child has $n^2$ candies, which are distributed in $n$ boxes. An operation consists in choosing two boxes that together contain an even number of candies and redistribute the candy from those boxes so that both contain the same amount of candy. Determine all the values of $n$ for which the child, after some operations, can get each box containing $n$ candies, no matter which the initial distribution of candies is.
In acute $\triangle ABC,$ let $D$ be the foot of perpendicular from $A$ on $BC$. Consider points $K, L, M$ on segment $AD$ such that $AK= KL= LM= MD$. Suppose the sum of the areas of the shaded region equals the sum of the areas of the unshaded regions in the following picture. Prove that $BD= DC$.
[img]http://s27.postimg.org/a0d0plr4z/Untitled.png[/img]
Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$
Given convex hexagon $ABCDEF$, inscribed in the circle. Prove that
$AC*BD*DE*CE*EA*FB \geq 27 AB * BC * CD * DE * EF * FA$
Let $n$ be a positive integer such that the sum of its positive divisors is at least $2022n$. Prove that $n$ has at least $2022$ distinct prime factors.
$n$ integers are arranged along a circle in such a way that each number is equal to the absolute value of the difference of the two numbers following that number in clockwise direction. If the sum of all numbers is $278$, how many different values can $n$ take?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 139
\qquad\textbf{(E)}\ \text{None of the above}
$
Let $m,n$ be positive integers and $p$ be a prime number.
Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.
[b]5/1/32.[/b] Find all pairs of rational numbers $(a, b)$ such that $0 < a < b$ and $a^a = b^b$.
The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $
A car goes along a straight highway at the speed of $60$ km per hour. A $100$ meter long fence is standing parallel to the highway. Every second, the passenger of the car measures the angle of vision of the fence. Prove that the sum of all angles measured by him is less than $1100$ degrees.
Determine whether exists positive integer $n$ such that the number $A=8^n+47$ is prime.
The RTP tax is a function $f(x)$, where $x$ is the total of the annual profits (in pesetas). Knowing that:
a) $f(x)$ is a continuous function
b) The derivative $\frac{df(x)}{dx}$ on the interval $0 \leq 6000$ is constant and equals zero; in the interval $6000< x < P$ is constant and equals $1$; and when $x>P$ is constant and equal 0.14.
c) $f(0)=0$ and $f(140000)=14000$.
Determine the value of the amount $P$ (in pesetas) and represent graphically the function $y=f(x)$.
Given real numbers $a_i,b_i~(i=1,2,\ldots,n)$ such that
\begin{align*}
&a_1\ge a_2\ge\ldots\ge a_n>0,\\
&b_1\ge a_1,\\
&b_1b_2\ge a_1a_2,\\
&\vdots\\
&b_1b_2\cdots b_n\ge a_1a_2\cdots a_n,
\end{align*}prove that $b_1+b_2+\ldots+b_n\ge a_1+a_2+\ldots+a_n$.
An infinite number of participants gathered for the Olympiad, who were registered under the numbers $1, 2,\ldots$. It turns out that for every $n = 1, 2,\ldots$ a participant with number $n{}$ has at least $n{}$ friends among the remaining participants (note: friendship is mutual). There is a hotel with an infinite number of double rooms. Prove that the participants can be accommodated in double rooms so that there is a couple of friends in each room.
[i]Proposed by V. Bragin, P. Kozhevnikov[/i]
A solid in Euclidean $3$-space extends from $z = \frac{-h}{2}$ to $z = \frac{+h}{2}$ and the area of the section $z = k$ is a polynomial in $k$ of degree at most $3$. Show that the volume of the solid is $\frac{h(B + 4M + T)}{6},$ where $B$ is the area of the bottom $(z = \frac{-h}{2})$, $M$ is the area of the middle section $(z = 0),$ and $T$ is the area of the top $(z = \frac{h}{2})$. Derive the formulae for the volumes of a cone and a sphere.
A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false?
(1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$.
(2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.
On a windless day, a pigeon can fly from Albatrocity to Finchester and back in $3$ hours and $45$ minutes. However, when there is a $10$ mile per hour win blowing from Albatrocity to Finchester, it takes the pigeon $4$ hours to make the round trip. How many miles is it from Albatrocity to Finchester?
In the triangle $ABC$, angle bisector $CD$ intersects the circumcircle of $ABC$ at the point $K$.
(a) Prove the equalities:
$$\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1$$where $I$ is the center of the inscribed circle of triangle $ABC$.
(b) On the segment $CK$ some point $P$ is chosen whose projections on $AC,BC,AB$ respectively are $P_1,P_2,P_3$. The lines $PP_3$ and $P_1P_2$ intersect at a point $M$. Find the locus of $M$ when $P$ moves around segment $CK$.
Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that
[b](a)[/b] The sum of the elements of $A$ is $0,$
[b](b)[/b] For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$.
Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that
\[|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.\]
Where $|X|$ denote the numbers of the elements in set $X$.
$f(x)$ is an even function with period of $2$. If $f(x)=x^{\frac{1}{1000}}$ when $x\in[0,1]$, then the order of $f\left(\frac{98}{19}\right),f\left(\frac{101}{17}\right),f\left(\frac{104}{15}\right)$ is________(from small to large).
For natural numbers $n$ and $b$, let $V(n, b)$ denote the number of decompositions of $n$ into the product of integers each of which is greater than $b$: for example
$$36 = 6\times 6 = 4\times 9 = 3\times 3\times 4 = 3\times 12,$$
i.e. $V(36,2) = 5$. Prove that $V(n, b) < n/b$ for all $n$ and $b$.
(N.B. Vasiliev, Moscow)
There is a set of $50$ cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card [i]lies beautifully[/i] if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly $25$ cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be?
[i]K.A.Sukhov[/i]
Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .