Found problems: 85335
Find all (finite) increasing arithmetic progressions, consisting only of prime numbers, such that the number of terms is larger than the common difference.
Let $a, b, c, d$ be integers with $a>b>c>d>0$. Suppose that $ac+bd=(b+d+a-c)(b+d-a+c)$. Prove that $ab+cd$ is not prime.
Determine the number of times and the positions in which it appears $\frac12$ in the following sequence of fractions:
$$ \frac11, \frac21, \frac12 , \frac31 , \frac22 , \frac13 , \frac41,\frac32,\frac23,\frac14,..., \frac{1}{1992}$$
Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find
\[\lim_{n\to\infty}f(n)^{1/n}\]
Let $ P(x)$ be a polynomial such that the following inequalities are satisfied:
\[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\]
and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\]
Prove that for every positive natural number $ n,$ $ P(n)$ is positive.
Let $P(x)$ be a polynomial with integer coefficients such that
\[P(\sqrt{2}\sin x) = -P(\sqrt{2}\cos x)\]
for all real numbers $x$. What is the largest prime that must divide $P(2019)$?
There are $n$ ($n \ge 10$) cards with numbers $1, 2, \ldots, n$ lying in a row on a table, face down, so that the numbers on any adjacent cards differ by at least $5.$ Is it always enough to turn at most $n-5$ cards to determine which of the cards has number $n$? (It is not necessary to turn the card with number $n$.)
Let $d(n)$ be the number of positive divisors of a natural number $n$.Find all $k\in \mathbb{N}$ such that there exists $n\in \mathbb{N}$ with $d(n^2)/d(n)=k$.
An equilateral triangle with side length 20 is subdivided using parallels to its sides into \( 20^2 = 400 \) smaller equilateral triangles of side length 1. Some segments of length 1, which are edges of these small triangles, must be colored red in such a way that no small triangle has all three of its edges colored red. Determine the maximum number of segments of length 1 that can be colored red.
Consider an acute triangle $ABC$ in which $A_1, B_1,$ and $C_1$ are the feet of the altitudes from $A, B,$ and $C,$ respectively, and $H$ is the orthocenter. The perpendiculars from $H$ onto $A_1C_1$ and $A_1B_1$ intersect lines $AB$ and $AC$ at $P$ and $Q,$ respectively. Prove that the line perpendicular to $B_1C_1$ that passes through $A$ also contains the midpoint of the line segment $PQ$.