Found problems: 117
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.
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.
Let $f(x) = x^3 + 3x^2 - 1$ have roots $a,b,c$.
(a) Find the value of $a^3 + b^3 + c^3$
(b) Find all possible values of $a^2b + b^2c + c^2a$
The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$.
If $a$ is real number such that $x_1$ and $x_2$, $x_1\neq x_2$ , are real numbers and roots of equation $x_2-x+a=0$. Prove that $\mid {x_1}^2-{x_2}^2 \mid =1$ iff $\mid {x_1}^3-{x_2}^3 \mid =1$
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation
\[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
Consider two quadratic equations \begin{align*}x^2+ax+b&=0, \\ x^2+cx+d&=0,\end{align*} with real coefficients. Find necessary and sufficient conditions such that the first equation has (real) roots $x,x_1,$ the second $x,x_2$ and $x>0,x_1>x_2$.
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root?
The equation
\[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\]
is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
Get rid of irrationality in the denominator of a fraction $$\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+2}$$.
Consider the function $y(x)$ satisfying the differential equation $y'' = -(1+\sqrt{x})y$ with $y(0)=1$ and $y'(0)=0.$ Prove that $y(x)$ vanishes exactly once on the interval $0< x< \pi \slash 2,$ and find a positive lower bound for the zero.
Let $a$, $b$ and $m$ be three positive real numbers and $a>b$. Which of the numbers $A=\sqrt{a+m}-\sqrt{a}$ and $B=\sqrt{b+m}-\sqrt{b}$ is bigger:
Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.
Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$.
Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?
Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$?
Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that
$\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct.
$\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$
Source: 1976 Euclid Part A Problem 8
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Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is
$\textbf{(A) } 1 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } -7 \qquad \textbf{(E) } \text{none of these}$