Olympiad problems

1978 IMO Longlists, 25

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 $a - c \ge 0$.

2010 Contests, 2

Tags: function , rhombus , vieta , geometry

Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.

2014 AIME Problems, 5

Tags: function , algebra , polynomial , quadratic , vieta

Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.

2021 India National Olympiad, 2

Tags: integer root , cubic , algebra , vieta

Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$ has all the roots to be integers. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

2003 AMC 12-AHSME, 21

Tags: algebra , polynomial , vieta

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$

2008 Federal Competition For Advanced Students, Part 2, 2

Tags: algebra , polynomial , vieta , number theory

(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$? (b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?

2002 AMC 10, 10

Tags: quadratic , vieta

Suppose that $ a$ and $ b$ are are nonzero real numbers, and that the equation $ x^2\plus{}ax\plus{}b\equal{}0$ has solutions $ a$ and $ b$. Then the pair $ (a,b)$ is $ \textbf{(A)}\ (\minus{}2,1) \qquad \textbf{(B)}\ (\minus{}1,2) \qquad \textbf{(C)}\ (1,\minus{}2) \qquad \textbf{(D)}\ (2,\minus{}1) \qquad \textbf{(E)}\ (4,4)$

2008 AMC 10, 9

Tags: quadratic , vieta , algebra

A quadratic equation $ ax^2\minus{}2ax\plus{}b\equal{}0$ has two real solutions. What is the average of the solutions? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac{b}{a} \qquad \textbf{(D)}\ \frac{2b}{a} \qquad \textbf{(E)}\ \sqrt{2b\minus{}a}$

2005 India National Olympiad, 3

Tags: quadratic , vieta , algebra

Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say $\alpha$. Prove that $a)$ $\alpha$ is real and negative; $b)$ the remaining third quadratic equation has non-real roots.

2006 Flanders Math Olympiad, 1

Tags: trigonometry , algebra , polynomial , vieta , complex number

(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 $d$.

1997 China Team Selection Test, 1

Tags: polynomial , inequalities , vieta , induction , algebra

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

2014 Contests, 1

Tags: vieta

If $\alpha$ and $\beta$ are the roots of the equation $3x^2+x-1=0$, where $\alpha>\beta$, find the value of $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$. $ \textbf{(A) }\frac{7}{9}\qquad\textbf{(B) }-\frac{7}{9}\qquad\textbf{(C) }\frac{7}{3}\qquad\textbf{(D) }-\frac{7}{3}\qquad\textbf{(E) }-\frac{1}{9} $

2016 Indonesia TST, 2

Tags: invariant , symmetry , quadratic , algebra , polynomial , vieta , number theory

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

1985 IberoAmerican, 1

Tags: polynomial , vieta , modular arithmetic , symmetry , algebra

Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]

2005 National Olympiad First Round, 15

Tags: function , vieta

For how many positive real numbers $a$ has the equation $a^2x^2 + ax+1-7a^2 = 0$ two distinct integer roots? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of above} $

2014 Cezar Ivănescu, 3

Tags: vieta , linear algebra

Let $ A,B,C,D $ be four $ 2\times 2 $ complex matrices such that $ A-D $ is invertible and such that $$ A^2+BA+C=0=D^2+BD+C. $$ Prove that $ \text{tr} (A+D) =-\text{tr} B $ and $ \det (AD) =\det C. $

1966 AMC 12/AHSME, 30

Tags: vieta

If three of the roots of $x^4+ax^2+bx+c=0$ are $1$, $2$, and $3$, then the value of $a+c$ is: $\text{(A)}\ 35 \qquad \text{(B)}\ 24\qquad \text{(C)}\ -12\qquad \text{(D)}\ -61 \qquad \text{(E)}\ -63$

2010 AMC 12/AHSME, 6

Tags: algebra , polynomial , vieta

A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

2010 Math Prize For Girls Problems, 20

Tags: trigonometry , algebra , polynomial , vieta , function , ratio , complex number

What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \] where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?

2021 Science ON all problems, 3

Tags: complex number , newton sums , vieta , algebra

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

2004 AMC 12/AHSME, 23

Tags: algebra , polynomial , vieta

The polynomial $ x^3\minus{}2004x^2\plus{}mx\plus{}n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $ n$ are possible? $ \textbf{(A)}\ 250,\!000 \qquad \textbf{(B)}\ 250,\!250 \qquad \textbf{(C)}\ 250,\!500 \qquad \textbf{(D)}\ 250,\!750 \qquad \textbf{(E)}\ 251,\!000$

2014 NIMO Problems, 5

Tags: algebra , polynomial , vieta , search

Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Justin Stevens[/i]

2012 ELMO Shortlist, 6

Tags: floor function , algebra , polynomial , vieta , quadratic , inequalities , number theory

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

2010 Romania National Olympiad, 1

Tags: quadratic , vieta , function , algebra

Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that \[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\] Prove that $(a_n)_{n\ge0}$ is a geometric sequence. [i]Lucian Dragomir[/i]

2006 Princeton University Math Competition, 3

Tags: algebra , polynomial , vieta

Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r^2_1+\dots+r^2_5$?