Found problems: 1148
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
The last digit of the number $x^2 +xy+y^2$ is zero (where $x$ and $y$ are positive integers). Prove that two last digits of this numbers are zeros.
Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ?
[i]A. Khrabrov[/i]
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.
Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.
If the sum of all the angles except one of a convex polygon is $ 2190^{\circ}$, then the number of sides of the polygon must be
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 15 \qquad
\textbf{(C)}\ 17 \qquad
\textbf{(D)}\ 19 \qquad
\textbf{(E)}\ 21$
Find the sum of the absolute values of the roots of $x^4 - 4x^3 - 4x^2 + 16x - 8 = 0$.
Let $a,b$ be the roots of the equation $x^2-px+q = 0$ and let $c, d$ be the roots of the equation $x^2 - rx + s = 0$, where $p, q, r,s$ are some positive real numbers. Suppose that $M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2}$ is an integer. Determine $a, b, c, d$.
Find an ordered pair $(a,b)$ of real numbers for which $x^2+ax+b$ has a non-real root whose cube is $343$.
Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.
$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.
Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.
Determine all positive integer $a$ such that the equation $2x^2 - 30x + a = 0$ has two prime roots, i.e. both roots are prime numbers.
Show that there do not exist any distinct natural numbers $ a$, $ b$, $ c$, $ d$ such that $ a^3\plus{}b^3\equal{}c^3\plus{}d^3$ and $ a\plus{}b\equal{}c\plus{}d$.
A sequence of three real numbers forms an arithmetic progression with a first term of $ 9$. If $ 2$ is added to the second term and $ 20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 81$
What quadratic polynomial whose coefficient of $x^2$ is $1$ has roots which are the complex conjugates of the solutions of $x^2 -6x+ 11 = 2xi-10i$? (Note that the complex conjugate of $a+bi$ is $a-bi$, where a and b are real numbers.)
For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.
Let p(x) be the polynomial $x^3 + 14x^2 - 2x + 1$. Let $p^n(x)$ denote $p(p^(n-1)(x))$. Show that there is an integer N such that $p^N(x) - x$ is divisible by 101 for all integers x.
Do there exist three different quadratic trinomials $f(x), g(x), h(x)$ such that the roots of the equation $f(x)=g(x)$ are $1$ and $4$, the roots of the equation $g(x)=h(x)$ are $2$ and $5$, and the roots of the equation $h(x)=f(x)$ are $3$ and $6$?
[i]Proposed by A. Golovanov[/i]
Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$?
$ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$
$\textbf{(B)}\ \text{The discriminant is 9}\qquad$
$\textbf{(C)}\ \text{The roots are imaginary}\qquad$
$\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$
$\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?