Olympiad problems

2016 NIMO Problems, 4

Tags: number theory , algebra , Dice , NIMO , probability , Quadratic

A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2-4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m+n$. [i] Proposed by Justin Stevens [/i]

1959 AMC 12/AHSME, 34

Tags: Vieta , irrational number , complex numbers , imaginary numbers , AMC , Quadratic , AMC 12

Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is: $ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$ $\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$

2011 All-Russian Olympiad, 2

Tags: quadratics , algebra proposed , algebra , Quadratic

Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?

2010 Dutch BxMO TST, 5

Tags: permutation , Quadratic , number theory , Perfect Square

For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.

1984 IMO Longlists, 13

Tags: number theory , Diophantine equation , Quadratic , polynomial , equation , IMO Shortlist

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

1984 IMO Shortlist, 2

Tags: number theory , Diophantine equation , Quadratic , polynomial , equation , IMO Shortlist

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

2000 Moldova National Olympiad, Problem 1

Tags: Quadratic , algebra , quadratics

Let $a,b,c$ be real numbers with $a,c\ne0$. Prove that if $r$ is a real root of $ax^2+bx+c=0$ and $s$ a real root of $-ax^2+bx+c=0$, then there is a root of a $\frac a2x^2+bx+c=0$ between $r$ and $s$.

2001 Slovenia National Olympiad, Problem 2

Tags: Quadratic

Find all rational numbers $r$ such that the equation $rx^2 + (r + 1)x + r = 1$ has integer solutions.

1964 Vietnam National Olympiad, 2

Tags: algebra , Quadratic , parametric equation

Draw the graph of the functions $y = | x^2 - 1 |$ and $y = x + | x^2 -1 |$. Find the number of roots of the equation $x + | x^2 - 1 | = k$, where $k$ is a real constant.

2010 Dutch BxMO TST, 5

Tags: permutation , Quadratic , number theory , Perfect Square

For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.

Estonia Open Junior - geometry, 2016.2.4

Tags: conics , parabola , area , Quadratic , algebra , analytic geometry

Let $d$ be a positive number. On the parabola, whose equation has the coefficient $1$ at the quadratic term, points $A, B$ and $C$ are chosen in such a way that the difference of the $x$-coordinates of points $A$ and $B$ is $d$ and the difference of the $x$-coordinates of points $B$ and $C$ is also $d$. Find the area of the triangle $ABC$.

2017 Tuymaada Olympiad, 5

Tags: Quadratic , trinomial , Integer Polynomial , algebra

Does there exist a quadratic trinomial $f(x)$ such that $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients are integers? (A. Khrabrov)

2004 Austria Beginners' Competition, 3

Tags: algebra , polynomial , sum of roots , Quadratic

Determine the value of the parameter $m$ such that the equation $(m-2)x^2+(m^2-4m+3)x-(6m^2-2)=0$ has real solutions, and the sum of the third powers of these solutions is equal to zero.

2007 Junior Balkan Team Selection Tests - Moldova, 6

Tags: angles , trinomial , Quadratic

The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$

2017 Romanian Master of Mathematics, 4

Tags: symmetry , graph , function , Quadratic , RMM

In the Cartesian plane, let $G_1$ and $G_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2 + q_1x + r_1$ and $f_2(x) = p_2x^2 + q_2x + r_2$, where $p_1 > 0 > p_2$. The graphs $G_1$ and $G_2$ cross at distinct points $A$ and $B$. The four tangents to $G_1$ and $G_2$ at $A$ and $B$ form a convex quadrilateral which has an inscribed circle. Prove that the graphs $G_1$ and $G_2$ have the same axis of symmetry.

2017 Romanian Masters In Mathematics, 4

Tags: symmetry , graph , function , Quadratic , RMM

In the Cartesian plane, let $G_1$ and $G_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2 + q_1x + r_1$ and $f_2(x) = p_2x^2 + q_2x + r_2$, where $p_1 > 0 > p_2$. The graphs $G_1$ and $G_2$ cross at distinct points $A$ and $B$. The four tangents to $G_1$ and $G_2$ at $A$ and $B$ form a convex quadrilateral which has an inscribed circle. Prove that the graphs $G_1$ and $G_2$ have the same axis of symmetry.

1993 ITAMO, 2

Tags: quadratic trinomial , trinomial , Quadratic , number theory , Rational roots , rational , primes

Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

2001 Austria Beginners' Competition, 2

Tags: algebra , quadratic polynomial , Quadratic

Consider the quadratic equation $x^2-2mx-1=0$, where $m$ is an arbitrary real number. For what values of $m$ does the equation have two real solutions, such that the sum of their cubes is equal to eight times their sum.