Found problems: 236
[b](a)[/b] Prove that
\[\frac {x^{2}}{\left(x \minus{} 1\right)^{2}} \plus{} \frac {y^{2}}{\left(y \minus{} 1\right)^{2}} \plus{} \frac {z^{2}}{\left(z \minus{} 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.
[b](b)[/b] Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.
[i]Author: Walther Janous, Austria[/i]
Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.
The graph of the polynomial \[P(x) \equal{} x^5 \plus{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx \plus{} e\] has five distinct $ x$-intercepts, one of which is at $ (0,0)$. Which of the following coefficients cannot be zero?
$ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
For a given value of $k$ the product of the roots of \[ x^2-3kx+2k^2-1=0 \]
is $7$. The roots may be characterized as:
$ \textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad$
$\textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary} $
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers?
$ \textbf{(A)}\ 794\qquad
\textbf{(B)}\ 796\qquad
\textbf{(C)}\ 798\qquad
\textbf{(D)}\ 800\qquad
\textbf{(E)}\ 802$
Let $x,y,z$ be 3 different real numbers not equal to $0$ that satisfiying
$x^2-xy=y^2-yz=z^2-zx$.
Find all the values of $\frac{x}{z}+\frac{y}{x}+\frac{z}{y}$ and $(x+y+z)^3+9xyz$.
Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$
What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]
Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?
If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals:
$ \textbf{(A) }10\qquad\textbf{(B) }9 \qquad\textbf{(C) }2\qquad\textbf{(D) }-2\qquad\textbf{(E) }-9 $
Quadratic equation $ x^2\plus{}ax\plus{}b\plus{}1\equal{}0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2\plus{}b^2$ is not a prime.
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
The roots of $x^2+bx+c=0$ are both real and greater than $1$. Let $s=b+c+1$. Then $s:$
$ \textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad$ $\textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad $
$\textbf{(E)}\text{ must be between -1 and +1} $
For some real numbers $ a$ and $ b$, the equation
\[ 8x^3 \plus{} 4ax^2 \plus{} 2bx \plus{} a \equal{} 0
\]has three distinct positive roots. If the sum of the base-$ 2$ logarithms of the roots is $ 5$, what is the value of $ a$?
$ \textbf{(A)}\minus{}\!256 \qquad
\textbf{(B)}\minus{}\!64 \qquad
\textbf{(C)}\minus{}\!8 \qquad
\textbf{(D)}\ 64 \qquad
\textbf{(E)}\ 256$
$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$
$(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that
$(\text{i})$ $x$ and $y$ are relatively prime;
$(\text{ii})$ $x$ divides $y^2+m;$
$(\text{iii})$ $y$ divides $x^2+m.$
Consider a polynomial $P(x) = ax^2 + bx + c$ with $a > 0$ that has two real roots $x_1, x_2$. Prove that the absolute values of both roots are less than or equal to $1$ if and only if $a + b + c \ge 0, a -b + c \ge 0$, and $a - c \ge 0$.
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 $n$
(a) Solve for $\theta\in\mathbb{R}$: $\cos(4\theta) = \cos(3\theta)$
(b) $\cos\left(\frac{2\pi}{7}\right)$, $\cos\left(\frac{4\pi}{7}\right)$ and $\cos\left(\frac{6\pi}{7}\right)$ are the roots of an equation of the form $ax^3+bx^2+cx+d = 0$ where $a, b, c, d$ are integers. Determine $a, b, c$ and $d$.
The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$.
Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$.
Find the minimum area of the figure bounded by the tangent tlines $ l_1,\ l_2$ and the curve $ y \equal{} f(x)$ .
Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.
Given that three roots of $f(x)=x^4+ax^2+bx+c$ are $2$, $-3$, and $5$, what is the value of $a+b+c$?