Found problems: 5802
The polynomials $ P(x)$ and $ Q(x)$ are given. It is known that for a certain polynomial $ R(x, y)$ the identity $ P(x) \minus{} P(y) \equal{} R(x, y) (Q(x) \minus{} Q(y))$ applies. Prove that there is a polynomial $ S(x)$ so that $ P(x) \equal{} S(Q(x)) \quad \forall x.$
Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d$ be the number of diagonals Alice drew. Find the least possible value that $d$ can obtain.
Prove that there exist infinitely many positive integers $n$ such that $3^n+2$ and $5^n+2$ are all composite numbers.
A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.
In a board school, there are 9 subjects, 512 students, and 256 rooms (two people in each room.) For every student there is a set (a subset of the 9 subjects) of subjects the student is interested in. Each student has a different set of subjects, (s)he is interested in, from all other students. (Exactly one student has no subjects (s)he is interested in.)
Prove that the whole school can line up in a circle in such a way that every pair of the roommates has the two people standing next to each other, and those pairs of students standing next to each other that are not roommates, have the following properties. One of the two students is interested in all the subjects that the other student is interested in, and also exactly one more subject.
A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.
Given two integers $h \geq 1$ and $p \geq 2$, determine the minimum number of pairs of opponents an $hp$-member parliament may have, if in every partition of the parliament into $h$ houses of $p$ member each, some house contains at least one pair of opponents.
Let $ x_1 , x_2, x_3 ...$ a succession of positive integers such that for every couple of positive integers $(m,n)$ we have $ x_{mn} \neq x_{m(n+1)}$ . Prove that there exists a positive integer $i$ such that $x_i \ge 2017 $.
Hi everybody!
I've an interesting problem!
Can you solve it?
Prove [b]Erdös-Ginzburg-Ziv Theorem[/b]: [i]"Among any $2n-1$ integers, there are some $n$ whose sum is divisible by $n$."[/i]
Find all natural numbers $n$ for which the following $5$ conditions hold:
$(1)$ $n$ is not divisible by any perfect square bigger than $1$.
$(2)$ $n$ has exactly one prime divisor of the form $4k+3$, $k\in \mathbb{N}_0$.
$(3)$ Denote by $S(n)$ the sum of digits of $n$ and $d(n)$ as the number of positive divisors of $n$. Then we have that $S(n)+2=d(n)$.
$(4)$ $n+3$ is a perfect square.
$(5)$ $n$ does not have a prime divisor which has $4$ or more digits.
A white equilateral triangle is split into $n^2$ equal smaller triangles by lines that are parallel to the sides of the triangle. Denote a [i]line of triangles[/i] to be all triangles that are placed between two adjacent parallel lines that forms the grid. In particular, a triangle in a corner is also considered to be a line of triangles.
We are to paint all triangles black by a sequence of operations of the following kind: choose a line of triangles that contains at least one white triangle and paint this line black (a possible situation with $n=6$ after four operations is shown in Figure 1; arrows show possible next operations in this situation). Find the smallest and largest possible number of operations.
Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and
\[2^ap^b=(p+2)^c-1.\]
Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: \[ (a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 . \]
[i]Viorel Vajaitu[/i]
Consider an odd prime $p$ and a positive integer $N < 50p$. Let $a_1, a_2, \ldots , a_N$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100}N$ times and $a_1 + a_2 + \cdots· + a_N$ is not divisible by $p$. Prove that there exists a permutation $b_1, b_2, \ldots , b_N$ of the $a_i$ such that, for all $k = 1, 2, \ldots , N$, the sum $b_1 + b_2 + \cdots + b_k$ is not divisible by $p$.
[i]Will Steinberg, United Kingdom[/i]
Let $a$ and $b$ be distinct positive integers such that $3^a + 2$ is divisible by $3^b + 2$. Prove that $a > b^2$.
[i]Proposed by Tynyshbek Anuarbekov, Kazakhstan[/i]
Prove that, if every three consecutive vertices of a convex $n{}$-gon, $n\geqslant 4$, span a triangle of area at least 1, then the area of the $n{}$-gon is (strictly) greater than $(n\log_2 n)/4-1/2.$
[i]Radu Bumbăcea & Călin Popescu[/i]
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose
\[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\]
Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?
We say that we extend a finite sequence of positive integers $(a_1,\dotsc,a_n)$ if we replace it by
\[(1,2,\dotsc,a_1-1,a_1,1,2,\dotsc,a_2-1,a_2,1,2,\dotsc,a_3-1,a_3,\dotsc,1,2,\dotsc,a_n-1,a_n)\]
i.e., each element $k$ of the original sequence is replaced by $1,2,\dotsc,k$. Géza takes the sequence $(1,2,\dotsc,9)$
and he extends it $2017$ times. Then he chooses randomly one element of the resulting sequence. What is the
probability that the chosen element is $1$?
Prove that $\sqrt {1+\sqrt {2+\ldots +\sqrt {n}}}<2$, $\forall n\ge 1$.
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying the following:
$i)$ $f(1)=1$;
$ii)$ $f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))$ for all $m,n \in \mathbb{Z}$.
Prove that for each $n\geq 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a,b\in S$.
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.
Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial \[ \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}. \]
Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $$2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1.$$
$\textbf{a)}$ Prove that the given sequence is an arithmetic progression.
$\textbf{b)}$ If $\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N},$ prove that every term of the sequence is an integer.