Practice (Intermediate)

3939

Let integers

$u$ and

$v$ be two integral roots to the quadratic equation

$x^2 + bx+c=0$ where

$b+c=298$ . If

$u < v$ , find the smallest possible value of

$v-u$ .

3940

Find all the ordered integers

$(a, b, c)$ which satisfy

$a+b+c=450$ and

$\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}=2c$ .

3942

$n$ straight lines are drawn on a plane such in such a way that no two of them are parallel and no three of them meet at one point. Show that the number of regions in which these lines divide the plane is

$\frac{(n)(n+1)}{2}+1$ .

3944

Show

$\frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$

3951

Find the value of the following expression:

$\binom{2020}{0}-\binom{2020}{2}+\binom{2020}{4}-\cdots+\binom{2020}{2020}$

3952

Given integers $a$ , $b$ , $n$ . Show that there exist integers $x$ , $y$ , such that $(a^2+b^2)^n = x^2 + y^2$ .

3954

The points $(0,0)$ , $(a,11)$ , and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$ .

3955

Find

$c$ if

$a$ ,

$b$ , and

$c$ and positive integers which satisfy

$c = (a+bi)^3 - 107i$

3957

Let

$z_1$ ,

$z_2$ ,

$z_3$ be complex numbers with nonzero imaginary parts such that

$|z_1| = |z_2| = |z_3|$ . Show that if

$z_1+z_2z_3$ ,

$z_2+z_1z_3$ ,

$z_3+z_1z_2$ are real, then

$z_1z_2z_3 = 1$ .

3961

The roots of

$x^2+ax+b+1$ are positive integers. Show that

$a^2+b^2$ is not a prime number.

3962

Let

$\alpha$ and

$\beta$ be the roots of

$x^2+px+1$ , and let

$\gamma$ and

$\sigma$ be the roots of

$x^2+qx+1$ . Show

$(\alpha - \gamma)(\beta-\gamma)(\alpha+\sigma)(\beta+\sigma) = q^2 - p^2$

3963

Let

$a$ ,

$b$ ,

$c$ be distinct real numbers. Show that there is a real number

$x$ such that

$x^2+2(a+b+c)x+3(ab+bc+ac)$ is negative.

3964

Consider the quadratic equation

$ax^2-bx+c=0$ where

$a$ ,

$b$ ,

$c$ are real numbers and

$a \ne 0$ . Find the values of

$a$ ,

$b$ ,

$c$ such that

$a$ and

$b$ are the roots of the equation and

$c$ is it's discriminant.

3965

Let

$b \ge 0$ be a real number. The product of the four real roots of the equations

$x^2+2bx+c=0$ and

$x^2+2cx+b=0$ is equal to

$1$ . Find the values of

$b$ and

$c$ .

3966

Solve the equation

$x^4-97x^3+2012x^2-97x+1=0$

3967

Show that if

$a$ ,

$b$ ,

$c$ are the lengths of the sides of a triangle, then the equation

$b^2x^2+(b^2+c^2-a^2)x + c^2=0$ does not have any real roots.

3968

Solve the equation $\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$

3970

Find all real numbers $m$ such that $x^2+my^2-4my+6y-6x+2m+8 \ge 0$ for every pair of real numbers $x$ and $y$ .

3971

Does there exist a polynomial

$P(x)$ such that

$P(1)=2015$ and

$P(2015)=2016$ ?

3972

Show that

$a=1+\sqrt{2}$ is irrational using the following steps: (a) Find a polynomial with integer coefficients that has

$a$ as a root. (b) Use the Rational Root Theorem to show

$a$ is irrational. Show that

$\sqrt{2}+\sqrt{3}$ is irrational using the same steps.

3973

Find

$a$ ,

$b$ , so that

$(x-1)^2$ divides

$ax^4+bx^3+1$ .

3975

Show that for each integer

$n$ the polynomial

$(\cos\theta+x\sin\theta)^n-\cos n\theta-x\sin n\theta$ is divisible by

$x^2+1$

3976

Show that

$\sum_{i=0}^n \binom{n}{i}^2 = \binom{2n}{n}$ .

3977

Let

$p(x) = x^3-3x+1$ . Show that if a complex number

$a$ is a root of

$p(x)$ , then

$a^2-2$ is also a root.

3979

Find the minimal value of

$\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8}$ .

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