Olympiad problems

1971 IMO Shortlist, 1

Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds: \[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\] Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$ \[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]

2017 Pakistan TST, Problem 3

Tags: algebraic identities , functional equation , algebra

Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$ $f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$

2015 Caucasus Mathematical Olympiad, 2

Tags: algebraic identities , identity , algebra

Vasya chose a certain number $x$ and calculated the following: $a_1=1+x^2+x^3, a_2=1+x^3+x^4, a_3=1+x^4+x^5, ..., a_n=1+x^{n+1}+x^{n+2} ,...$ It turned out that $a_2^2 = a_1a_3$. Prove that for all $n\ge 3$, the equality $a_n^2 = a_{n-1}a_{n+1}$ holds.

1969 IMO Shortlist, 65

Tags: algebraic identities , algebra

$(USS 2)$ Prove that for $a > b^2,$ the identity ${\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}$

1998 IMO Shortlist, 2

Tags: number theory , algebra , algebraic identities , floor function

Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor $ for all positive integers $n$. (Note that $\lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$.)

1967 IMO Longlists, 57

Tags: sequence , algebraic identities , algebra

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

2016 Irish Math Olympiad, 8

Tags: algebraic identities , system of equations , algebra

Suppose $a, b, c$ are real numbers such that $abc \ne 0$. Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$. Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$.

1971 IMO Longlists, 5

Tags: recurrence relation , polynomial , algebraic identities , algebra

Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds: \[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\] Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$ \[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]

1967 IMO Shortlist, 1

Tags: sequence , algebraic identities , algebra

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

1969 IMO Longlists, 65

Tags: algebra , algebraic identities

$(USS 2)$ Prove that for $a > b^2,$ the identity ${\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}$

2012 India Regional Mathematical Olympiad, 6

Tags: algebra , identity , algebraic identities

Show that for all real numbers $x,y,z$ such that $x + y + z = 0$ and $xy + yz + zx = -3$, the expression $x^3y + y^3z + z^3x$ is a constant.

1967 IMO, 5

Tags: algebra , sequence , algebraic identities

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$