Practice (90/1000)

3937

Does the expression $x+\sqrt{2x^2-2x+1}$ has either maximum or minimal value?

3938

In trapezoid $ABCD$, $AD\parallel BC$ and $AD:BC=1:2$. Point $F$ lies on $AB$ and point $E$ is on $CF$. If $S_{\triangle{AOF}}:S_{\triangle{DOE}}=1:3$ and $S_{\triangle{BEF}}=24$, find the area of $\triangle{AOF}$.

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

3943

The Fibonacci sequence $(F_n)_{n\ge 0}$ is defined by the recurrence relation $F_{n+2}=F_{n+1}+F_{n}$ with $F_{0}=0$ and $F_{1}=1$. Prove that for any $m$, $n \in \mathbb{N}$, we have $$F_{m+n+1}=F_{m+1}F_{n+1}+F_{m}F_{n}.$$ Deduce from here that $F_{2n+1}=F^2_{n+1}+F^2_{n}$ for any $n \in \mathbb{N}$

3944

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

3947

Let $f$ be a function from $\mathbb{N}$ to $\mathbb{N}$ such that

(i) $f(1)=0$

(ii) $f(2n)=2f(n)+1)$

(iii) $f(2x+1)=2f(n)$

Find the least value of $n$ such that $f(n)=2016$.

3949

Let $z=\cos{\theta} + i\sin{\theta} $. Show $z^{-1} = \cos{\theta} - i\sin{\theta}$.

3950

Compute $\sqrt{i}$.

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 $$(a^2+b^2)^n = x^2 + y^2$$.

3953

Solve the equation $z^4+1=0$.

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

3956

Find the number of ordered pairs $(a, b)$ of real numbers such that $$(a+bi)^{2016}=a-bi$$.

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 $z_1z_2z_3 = 1$.

3958

Let $(x^{2017}+x^{2019}+2)^{2018} = a_0+a_1x+\cdots+a_nx^n$. Find $$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$

3959

The sum and product of two numbers are equal to $y$. For which values of $y$ are these two numbers real?

3960

Let $m$ and $n$ be the roots of $P(x)=ax^2+bx+c$. Find the coefficients of the quadratic polynomial whose roots are $m^2-n$ and $n^2-m$.

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 $c$.

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