Found problems: 1148
Consider all the numbers of the form $x+yt+zt^2$, with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$. Prove that if $x+yt+zt^2\not= 0$, then there exist rational numbers $u,v,w$ such that
\[(x+yt+z^2)(u+vt+wt^2)=1\]
The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$
For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$
The sum $$\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\cdots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}$$ is a root of the quadratic $x^2+x+c$. What is $c$ in terms of $n$?
$\text{(A) }-\frac{n}{2}\qquad\text{(B) }2n\qquad\text{(C) }-2n\qquad\text{(D) }n+\frac{1}{2}\qquad\text{(E) }n-2$
Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition
\[y^2 \equiv x^3 - x \pmod p\]
is exactly $p.$
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$.
Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that:
\[mp=x_1^2+x_2^2+x_3^2.\]
Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$. It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$
Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$.
(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)
Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.
Find the integer numbers $a, b, c$ such that the function $f: R \to R$, $f(x) = ax^2 +bx + c$ satisfies the equalities : $$f(f(1) ))= f (f(2 ) )= f(f (3 ))$$
We may say concerning the solution of
\[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0
\]
that:
$ \textbf{(A)}\ \text{there is only one root}\qquad
\textbf{(B)}\ \text{the sum of the roots is }{\plus{}1}\qquad
\textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\
\textbf{(D)}\ \text{the product of the roots is }{\plus{}4}\qquad
\textbf{(E)}\ \text{the product of the roots is }{\minus{}6}$
Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.
A function $ f(x)$ is [i]periodic[/i] if there is a positive number $ p$ such that $ f(x\plus{}p) \equal{} f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6$, $BC=5=DA$, and $CD=4$. Draw circles of radius 3 centered at $A$ and $B$, and circles of radius 2 centered at $C$ and $D$. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p$, where $k$, $m$, $n$, and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p$.
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
[i]Proposed by Ahaan S. Rungta[/i]
For what real values of $ K$ does $ x \equal{} K^2 (x\minus{}1)(x\minus{}2)$ have real roots?
$ \textbf{(A)}\ \text{none} \qquad
\textbf{(B)}\ \minus{}2<K<1 \qquad
\textbf{(C)}\ \minus{}2 \sqrt{2} < K < 2 \sqrt{2} \qquad
\textbf{(D)}\ K>1 \text{ or } K<\minus{}2 \qquad
\textbf{(E)}\ \text{all}$
Let $f(x)=x^2+bx+c$, where $b,c \in \mathbb{R}$ and $b>0$
Do there exist disjoint sets $A$ and $B$, whose union is $[0,1]$ and $f(A)=B$, where $f(X)=\{f(x), x \in X\}$
[i]D. Zmiaikou[/i]
Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.
A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial
The following sentece is written on a board:
[center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center]
Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?
Solve the equation
\[\int_0^1(x+y)^2u(x)dx=\lambda u(y)+1\]
Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}