Found problems: 85335
Last year $100$ adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was $4$. What was the total number of cats and kittens received by the shelter last year?
$ \textbf{(A)}\ 150
\qquad\textbf{(B)}\ 200
\qquad\textbf{(C)}\ 250
\qquad\textbf{(D)}\ 300
\qquad\textbf{(E)}\ 400
$
Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.
Are there infinitely many natural numbers $n$ such that the sum of 2019th powers of the digits of $n$ is
equal to $n$ ? [b]You don't need to find any such $n$. Just provide mathematical justi fication if you
think there are in finitely many or finitely many such natural numbers[/b]
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
Let $T_m$ be a number of non-congruent triangles which perimeter is $m$ and all its sides are positive integers. Prove that:
$a)$ $T_{1999} > T_{2000}$
$b)$ $T_{4n+1}=T_{4n-2}+n$, $(n \in \mathbb{N})$
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.
Given a regular nonagon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$ with side length $1$. Diagonals $A_3A_7$ and $A_4A_8$ intersect at point $P$. Find the length of segment $P A_1$.
Positive real numbers $a,b,c$ are given such that $abc=1$. Prove that$$a+b+c+\frac{3}{ab+bc+ca}\geq4.$$
There is one nonzero digit in every cell of $2017\times 2017 $ table.
On the board we writes $4034$ numbers that are rows and columns of table. It is known, that $4033$ numbers are divisible by prime $p$ and last is not divisible by $p$. Find all possible values of $p$.
[hide=Example]Example for $2\times2$. If table is
|1|4|
|3|7|.
Then numbers on board are $14,37,13,47$[/hide]
Call a subset $A$ of the cyclic group $(\mathbb{Z}_n,+)$ [i]rich[/i] if for all $x,y \in \mathbb{Z}_n$ there exists $r \in \mathbb{Z}_n$ such that $x-r,x+r,y-r,y+r$ are all in $A$. For what $\alpha$ is there a constant $C_\alpha>0$ such that for each odd positive integer $n$, every rich subset $A \subset \mathbb{Z}_n$ has at least $C_\alpha n^\alpha$ elements?
Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$, and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \neq A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ and $(c_1)$ be the circumcircles of the triangles $\triangle AEZ$ and $\triangle BEZ$, respectively. Let $(c_2)$ be an arbitrary circle passing through the points $A$ and $E$. Suppose $(c_1)$ meets the line $CZ$ again at the point $F$, and meets $(c_2)$ again at the point $N$. If $P$ is the other point of intersection of $(c_2)$ with $AF$, prove that the points $N$, $B$, $P$ are collinear.
Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons?
by N.Beluhov
A convex quadrilateral $ABCD$ is given. Let $O_a$ be the circumcenter of the triangle $DBC$, and define $O_b,O_c$ and $O_d$ similarly. The points $O_a, O_b, O_c, O_d$ are the vertices of a convex quadrilateral. Prove that its area is equal to half of the absolute value of the difference between the areas of $AO_bCO_d$ and $BO_cDO_a$.
[i]Proposed by V. Dubrovsky[/i]
The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Which of the following triangles cannot exist?
$\textbf{(A)}\ \text{An acute isosceles triangle}$
$\textbf{(B)}\ \text{An isosceles right triangle}$
$\textbf{(C)}\ \text{An obtuse right triangle}$
$\textbf{(D)}\ \text{A scalene right triangle}$
$\textbf{(E)}\ \text{A scalene obtuse triangle}$
Show that an arbitary triangle can be dissected by straight line segments into three parts in three different ways so that each part has a line of symmetry.
Find the maximal possible numbers one can choose from $1,\ldots,100$ such that none of the products of non-empty subset of this numbers was a perfect square.
Given an acute and scalene triangle $ ABC$, let $ H$ be its orthocenter, $ O$ its circumcenter, $ E$ and $ F$ the feet of the altitudes drawn from $ B$ and $ C$, respectively. Line $ AO$ intersects the circumcircle of the triangle again at point $ G$ and segments $ FE$ and $ BC$ at points $ X$ and $ Y$ respectively. Let $ Z$ be the point of intersection of line $ AH$ and the tangent line to the circumcircle at $ G$. Prove that $ HX$ is parallel to $ YZ$.
Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that at least 85 dominoes(1×2 rectangle) can be removed from the remainder.
Proposed by S. Berlov
What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number?
$\textbf{(A) }108\qquad\textbf{(B) }120\qquad\textbf{(C) }126\qquad\textbf{(D) }132\qquad\textbf{(E) }144$
Prove that there does not exist a natural number which, upon transfer of its initial digit to the end, is increased five, six or eight times.
A regular $100$-gon was cut into several parallelograms and two triangles. Prove that these triangles are congruent.
In a tournament with $2015$ teams, each team plays every other team exactly once and no ties occur. Such a tournament is [i]imbalanced [/i]if for every group of $6$ teams, there exists either a team that wins against the other $5$ or a team that loses to the other $5$. If the teams are indistinguishable, what is the number of distinct imbalanced tournaments that can occur?
Prove that any group of people can be divided into two disjoint groups $A$ and $B$ such that any member from $A$ has at least half of his acquaintances in $B$ and any member from $B$ has at least half of his acquaintances in $A$ (acquaintance is reciprocal).
Point $P$ lies inside a triangle $ABC$. Let $D,E$ and $F$ be reflections of the point $P$ in the lines $BC,CA$ and $AB$, respectively. Prove that if the triangle $DEF$ is equilateral, then the lines $AD,BE$ and $CF$ intersect in a common point.