Olympiad problems

1976 IMO Shortlist, 9

Tags: algebra , polynomial , recurrence relation , roots , IMO , chebyshev polynomial , imo 1976

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

1940 Putnam, B5

Tags: polynomial , rational , algebra , roots

Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

1976 IMO, 2

Tags: algebra , polynomial , recurrence relation , roots , IMO , chebyshev polynomial , imo 1976

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

2020 Brazil Undergrad MO, Problem 6

Tags: function , college contests , quadratics , roots , Fixed point , Brazilian Undergrad MO , Brazilian Undergrad MO 2020

Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times). a) Find the number of distinct real roots of the equation $f^{3}(x) = x$ b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$

1985 IMO Shortlist, 11

Tags: algebra , polynomial , roots , algorithm , IMO Shortlist

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.

1966 IMO Shortlist, 35

Tags: algebra , polynomial , irrational number , roots , IMO Shortlist , IMO Longlist

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.

1967 IMO Longlists, 48

Tags: algebra , equation , roots , IMO Shortlist , IMO Longlist , Exponential equation

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

1967 IMO Longlists, 43

Tags: algebra , polynomial , Diophantine equation , parametric equation , roots , IMO Shortlist , IMO Longlist

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 $\lambda.$

1947 Putnam, B5

Tags: Putnam , Integer Polynomial , roots

Let $a,b,c,d$ be distinct integers such that $$(x-a)(x-b)(x-c)(x-d) -4=0$$ has an integer root $r.$ Show that $4r=a+b+c+d.$

2013 German National Olympiad, 6

Tags: Sequence , number theory , roots , real number

Define a sequence $(a_n)$ by $a_1 =1, a_2 =2,$ and $a_{k+2}=2a_{k+1}+a_k$ for all positive integers $k$. Determine all real numbers $\beta >0$ which satisfy the following conditions: (A) There are infinitely pairs of positive integers $(p,q)$ such that $\left| \frac{p}{q}- \sqrt{2} \, \right| < \frac{\beta}{q^2 }.$ (B) There are only finitely many pairs of positive integers $(p,q)$ with $\left| \frac{p}{q}- \sqrt{2} \,\right| < \frac{\beta}{q^2 }$ for which there is no index $k$ with $q=a_k.$

1970 Putnam, A2

Tags: Putnam , algebra , polynomial , roots

Consider the locus given by the real polynomial equation $$ Ax^2 +Bxy+Cy^2 +Dx^3 +E x^2 y +F xy^2 +G y^3=0,$$ where $B^2 -4AC <0.$ Prove that there is a positive number $\delta$ such that there are no points of the locus in the punctured disk $$0 <x^2 +y^2 < \delta^2.$$

1966 IMO Shortlist, 48

Tags: algebra , equation , number theory , roots , parametric equation , IMO Longlist , IMO Shortlist

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

2013 India Regional Mathematical Olympiad, 6

Tags: number theory , polynomial , roots , factorial , greatest common divisor

Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.

1955 Moscow Mathematical Olympiad, 314

Tags: polynomial , roots , algebra

Prove that the equation $x^n - a_1x^{n-1} - a_2x^{n-2} - ... -a_{n-1}x - a_n = 0$, where $a_1 \ge 0, a_2 \ge 0, . . . , a_n \ge 0$, cannot have two positive roots.

1953 Putnam, A7

Tags: Putnam , trigonometry , polynomial , roots

Assuming that the roots of $x^3 +px^2 +qx +r=0$ are all real and positive, find the relation between $p,q,r$ which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle.

1973 Putnam, A4

Tags: Putnam , function , roots , real analysis

How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?

1999 Bosnia and Herzegovina Team Selection Test, 6

Tags: algebra , polynomial , roots , Imaginary

It is given polynomial $$P(x)=x^4+3x^3+3x+p, (p \in \mathbb{R})$$ $a)$ Find $p$ such that there exists polynomial with imaginary root $x_1$ such that $\mid x_1 \mid =1$ and $2Re(x_1)=\frac{1}{2}\left(\sqrt{17}-3\right)$ $b)$ Find all other roots of polynomial $P$ $c)$ Prove that does not exist positive integer $n$ such that $x_1^n=1$

2020/2021 Tournament of Towns, P4

Tags: algebra , roots , Tournament of Towns

It is well-known that a quadratic equation has no more than 2 roots. Is it possible for the equation $\lfloor x^2\rfloor+px+q=0$ with $p\neq 0$ to have more than 100 roots? [i]Alexey Tolpygo[/i]

1970 Czech and Slovak Olympiad III A, 3

Tags: parameterization , algebra , roots , inequalities

Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]

1985 Spain Mathematical Olympiad, 7

Tags: polynomial , roots , Integer Polynomial , polynomial equation , algebra

Find the values of $p$ for which the equation $x^5 - px-1 = 0$ has two roots $r$ and $s$ which are the roots of equation $x^2-ax+b= 0$ for some integers $a,b$.

2020 German National Olympiad, 3

Tags: inequalities , inequalities proposed , polynomial , roots , algebra

Show that the equation \[x(x+1)(x+2)\dots (x+2020)-1=0\] has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies \[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\]

1996 German National Olympiad, 6a

Tags: polynomial , roots , algebra

Prove the following statement: If a polynomial $p(x) = x^3 + Ax^2 + Bx +C$ has three real positve roots at least two of which are distinct, then $A^2 +B^2 +18C > 0$.

2012 Dutch BxMO/EGMO TST, 1

Tags: trinomial , quadratics , polynomial , roots , algebra

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 $x = 7$?

2000 Bosnia and Herzegovina Team Selection Test, 1

Tags: algebra , roots , equation

Find real roots $x_1$, $x_2$ of equation $x^5-55x+21=0$, if we know $x_1\cdot x_2=1$

1967 IMO Shortlist, 2

Tags: algebra , polynomial , Diophantine equation , parametric equation , roots , IMO Shortlist , IMO Longlist

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 $\lambda.$