Found problems: 85335
Do there exist positive integers $x, y$, such that $x+y, x^2+y^2, x^3+y^3$ are all perfect squares?
Find the Betti numbers of the surface $x_1^2+\cdots+x_k^2-y_1^2-\cdots-y_l^2=1$ and the set $x_1^2+\cdots+x_k^2\le1+y_1^2+\cdots+y_l^2$ in a $(k+l)$-dimensional linear space.
Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$. Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$.
A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $\tfrac{p}{2}$ such that $\tfrac{ab-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are not mundane.
The Bernoulli sequence $\{B_{n}\}_{n \ge 0}$ is defined by \[B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)\] Show that for all $n \in \mathbb{N}$, \[(-1)^{n}B_{n}-\sum \frac{1}{p},\] is an integer where the summation is done over all primes $p$ such that $p| 2k-1$.
Let $ABC$ be a triangle such that $AB=2$, $BC=3$, and $AC=4$. A circle passing through $A$ intersects $AB$ at $D$, $AC$ at $E$, and $BC$ at $M$ and $N$ such that $BM=MN=NC$. Find $DE$.
Let $f(n) = \sum_{gcd(k,n)=1,1\le k\le n}k^3$ . If the prime factorization of $f(2020)$ can be written as $p^{e_1}_1 p^{e_2}_2 ... p^{e_k}_k$, find $\sum^k_{i=1} p_ie_i$.
The incircle $\omega$ of triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $D, E$ and $E$, respectively. Point $G$ lies on circle $\omega$ in such a way that $FG$ is a diameter. Lines $EG$ and $FD$ intersect at point $H$. Prove that $AB \parallel CH$.
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
Of the following expressions the one equal to $ \frac{a^{\minus{}1}b^{\minus{}1}}{a^{\minus{}3} \minus{} b^{\minus{}3}}$ is:
$ \textbf{(A)}\ \frac{a^2b^2}{b^2 \minus{} a^2}\qquad
\textbf{(B)}\ \frac{a^2b^2}{b^3 \minus{} a^3}\qquad
\textbf{(C)}\ \frac{ab}{b^3 \minus{} a^3}\qquad
\textbf{(D)}\ \frac{a^3 \minus{} b^3}{ab}\qquad
\textbf{(E)}\ \frac{a^2b^2}{a \minus{} b}$
There are $2000$ cities in the country, each of which has exactly three roads to other cities. Prove that you can close $1000$ roads, so that there is not a single closed route in the country, consisting of an odd number of roads.
Construct a quadrilateral given its side lengths and the length of the segment joining the midpoints of its diagonals.
(IZ Titovich)
We are going to colour the cells of a $2015 \times 2015$ board such that there are none of the following:
$1)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the right of the upper cell,
$2)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the left of the lower cell.
What is the minimum number of colours $k$ required to make such a colouring possible?
Determine all positive integers $n$ for which the equation \[x^{n}+(2+x)^{n}+(2-x)^{n}= 0\] has an integer as a solution.
Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes.
How many real pairs $(x,y)$ are there such that
\[
x^2+2y = 2xy \\
x^3+x^2y = y^2
\]
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{None}
$
$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half?
$ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$
Given a line segment $AB$ and a point $C$ lies inside it such that $AB=3 \cdot AC$ . Construct a parallelogram $ACDE$ such that $AC=DE=CE>AR$. Let $Z$ be a point on $AC$ such that $\angle AEZ=\angle ACE =\omega$. Prove that the line passing through point $B$ and perpendicular on side $EC$, and the line passing through point $D$ and perpendicular on side $AB$, intersect on point , let it be $K$, lying on line $EZ$.
There are a family of $5$ siblings. They have a pile of at least $2$ candies and are trying to split them up
amongst themselves. If the $2$ oldest siblings share the candy equally, they will have $1$ piece of candy left over.
If the $3$ oldest siblings share the candy equally, they will also have $1$ piece of candy left over. If all $5$ siblings
share the candy equally, they will also have $1$ piece left over. What is the minimum amount of candy required
for this to be true?
Find all functions ${f : (0, +\infty) \rightarrow R}$ satisfying $f(x) - f(x+ y) = f \left( \frac{x}{y}\right) f(x + y)$ for all $x, y > 0$.
(Austria)
Let be three positive real numbers $ x,y,z, $ whose product is $ 1. $ Prove that:
$$ \sum_{\text{cyc}} \frac{3}{\sqrt{1+x+xy}} \le \sqrt 3<3\sqrt 3\le \sum_{\text{cyc}} \sqrt{1+x+xy} $$
A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$?
(S . Fomin, Leningrad)
Let be three polynomials of degree two $ p_1,p_2,p_3\in\mathbb{R} [X] $ and the function
$$ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max\left( p_1(x),p_2(x),p_3(x)\right) . $$
Then, $ f $ is differentiable if and only if any of these three polynomials dominates the other two.
Does there exist a sequence $(a_n)_{n\in\mathbb N}$ of distinct positive integers such that:
(i) $a_n<1999n$ for all $n$;
(ii) none of the $a_n$ contains three decimal digits $1$?