Practice (90/1000)

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

3969

Find all pairs of integers $(x,y)$ such that $$x^3+y^3=(x+y)^2$$.

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

3974

Find all polynomials with integer coefficients $p(x)$ that satisfy the following identity $$2p(2x)=p(3x)+p(x)$$.

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.

3978

Given $$P(x)=(1+x+x^2)^{100}=a_0+a_1x+\cdots+a_{200}x^{200}$$

Compute the following sums:

$S_1=a_0+a_1+a_2+a_3 +\cdots+a_{200}$
$S_2=a_0+a_2+a_4+a_6 +\cdots+a_{200}$.

3979

Find the minimal value of $\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8}$.

4034

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?

4095

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

4096

Find all numbers $n \ge 3$ for which there exists real numbers $a_1, a_2, ..., a_{n+2}$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and\[a_{i}a_{i+1} + 1 = a_{i+2}\]for $i = 1, 2, ..., n.$

4100

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC \cdot DA.$ Point $X$ lies inside $ABCD$ so that $\angle XAB = \angle XCD$ and $\angle XBC = \angle XDA.$ Prove that $\angle BXA + \angle DXC = 180^{\circ}$ .

4101

Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find $S$.

4102

The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$.

4103

Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders $RRGGG$, $GGGGR$, or $RRRRR$ will make Kathy happy, but $RRRGR$ will not. Find the probability that Kathy will be happy.

4104

In $\triangle ABC, AB = AC = 10$ and $BC = 12$. Point $D$ lies strictly between $A$ and $B$ on $\overline{AB}$ and point $E$ lies strictly between $A$ and $C$ on $\overline{AC}$ so that $AD = DE = EC$. Then $AD$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

4105

For each ordered pair of real numbers $(x,y)$ satisfying\[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that\[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$.

4106

Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.

4107

A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).

back to index | new