Olympiad problems

2011 Princeton University Math Competition, A7 / B8

Tags: algebra

Let $\alpha_1,\alpha_2,\dots,\alpha_6$ be a fixed labeling of the complex roots of $x^6-1$. Find the number of permutations $\{\alpha_{i_1},\alpha_{i_2},\dots,\alpha_{i_6}\}$ of these roots such that if $P(\alpha_1, \dots, \alpha_6) = 0$, then $P(\alpha_{i_1},\dots,\alpha_{i_6}) = 0$, where $P$ is any polynomial with rational coefficients.

2017 Hanoi Open Mathematics Competitions, 1

Tags: polynomial , algebra , sum

Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 4x^2 -3x + 2$. The sum $|x_1| + |x_2| + |x_3|$ is (A): $4$ (B): $6$ (C): $8$ (D): $10$ (E): None of the above.

1999 Polish MO Finals, 1

Tags: algebra

For which $n$ do the equations have a solution in integers: \begin{eqnarray*}x_1 ^2 + x_2 ^2 + 50 &=& 16x_1 + 12x_2 \\ x_2 ^2 + x_3 ^2 + 50 &=& 16x_2 + 12x_3 \\ \cdots \quad \cdots \quad \cdots & \cdots & \cdots \quad \cdots \\ x_{n-1} ^2 + x_n ^2 + 50 &=& 16x_{n-1} + 12x_n \\ x_n ^2 + x_1 ^2 + 50 &=& 16x_n + 12x_1 \end{eqnarray*}

2009 Greece Team Selection Test, 3

Tags: algebra

Find all triples $(x,y,z)\in \mathbb{R}^{3}$ such that $x,y,z>3$ and $\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36$

2011 Puerto Rico Team Selection Test, 3

Tags: prime number , number theory , algebra

(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3. (b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.

2014 Iran MO (3rd Round), 3

Tags: algebra , polynomial , complex analysis

Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

2005 China Team Selection Test, 3

Tags: function , algebra

Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.

2017 Baltic Way, 2

Tags: inequalities , algebra , system of equations

Does there exist a finite set of real numbers such that their sum equals $2$, the sum of their squares equals $3$, the sum of their cubes equals $4$, ..., and the sum of their ninth powers equals $10$?

2016 CCA Math Bonanza, T8

Tags: polynomial , algebra

As $a$, $b$ and $c$ range over [i]all[/i] real numbers, let $m$ be the smallest possible value of $$2\left(a+b+c\right)^2+\left(ab-4\right)^2+\left(bc-4\right)^2+\left(ca-4\right)^2$$ and $n$ be the number of ordered triplets $\left(a,b,c\right)$ such that the above quantity is minimized. Compute $m+n$. [i]2016 CCA Math Bonanza Team #8[/i]

2014 Thailand Mathematical Olympiad, 2

Tags: function , functional equation , algebra

Find all functions $f : R \to R$ satisfying $f(xy - 1) + f(x)f(y) = 2xy - 1$ for all real numbers $x, y$

2003 Switzerland Team Selection Test, 7

Tags: polynomial , trinomial , quadratic polynomial , algebra

Find all polynomials $Q(x)= ax^2+bx+c$ with integer coefficients for which there exist three different prime numbers $p_1, p_2, p_3$ such that $|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11$.

2023 Purple Comet Problems, 18

Tags: algebra , floor function

For real number $x$, let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\}$ denote the fractional part of $x$, that is $\{x\} = x -\lfloor x\rfloor$. The sum of the solutions to the equation $2\lfloor x\rfloor^2 + 3\{x\}^2 = \frac74 x \lfloor x\rfloor$ can be written as $\frac{p}{q} $, where $p$ and $q$ are prime numbers. Find $10p + q$.

1986 Austrian-Polish Competition, 9

Tags: functional , algebra , continuous , composition , functional equation

Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.

1987 All Soviet Union Mathematical Olympiad, 462

Tags: inequalities , algebra

Prove that for every natural $n$ the following inequality is held: $$(2n + 1)^n \ge (2n)^n + (2n - 1)^n$$

2004 Junior Balkan Team Selection Tests - Moldova, 8

Tags: product , algebra

The positive real numbers $a$ and $b$ ($a> b$) are written on the board. At every step, with numbers written on the board, one of the following operations can be performed: a) choose one of the numbers and write its square or its inverse. b) choose two numbers written on the board ¸and write their sum or their positive difference. Show how the product $a \cdot b$ can be obtained with the help of the defined operations.

2013 Korea Junior Math Olympiad, 1

Tags: inequalities , algebra

Compare the magnitude of the following three numbers. $$ \sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}} $$

1902 Eotvos Mathematical Competition, 1

Tags: trinomial , algebra

Prove that any quadratic expression $$Q(x) = Ax^2 + Bx + C$$ (a) can be put into the form $$Q(x) = k \frac{x(x- 1)}{1 \cdot 2} + \ell x + m$$ where $k, \ell, m$ depend on the coefficients $A,B,C$ and (b) $Q(x)$ takes on integral values for every integer $x$ if and only if $k, \ell, m$ are integers.

2007 Bulgarian Autumn Math Competition, Problem 12.1

Tags: parameterization , parameter , trigonometric equation , algebra

Determine the values of the real parameter $a$, such that the equation \[\sin 2x\sin 4x-\sin x\sin 3x=a\] has a unique solution in the interval $[0,\pi)$.

1983 IMO Longlists, 24

Tags: algebra

Every $x, 0 \leq x \leq 1$, admits a unique representation $x = \sum_{j=0}^{\infty} a_j 2^{-j}$, where all the $a_j$ belong to $\{0, 1\}$ and infinitely many of them are $0$. If $b(0) = \frac{1+c}{2+c}, b(1) =\frac{1}{2+c},c > 0$, and \[f(x)=a_0 + \sum_{j=0}^{\infty}b(a_0) \cdots b(a_j) a_{j+1}\] show that $0 < f(x) -x < c$ for every $x, 0 < x < 1.$

2023 Kyiv City MO Round 1, Problem 1

Tags: expression , algebra

Find the integer which is closest to the value of the following expression: $$((7 + \sqrt{48})^{2023} + (7 - \sqrt{48})^{2023})^2 - ((7 + \sqrt{48})^{2023} - (7 - \sqrt{48})^{2023})^2$$

2022 Bulgarian Spring Math Competition, Problem 10.1

Tags: system of equations , inequality , algebra

If $x, y, z \in \mathbb{R}$ are solutions to the system of equations $$\begin{cases} x - y + z - 1 = 0\\ xy + 2z^2 - 6z + 1 = 0\\ \end{cases}$$ what is the greatest value of $(x - 1)^2 + (y + 1)^2$?

2023-IMOC, A6

Tags: inequalities , algebra

We define \[f(x,y,z)=|xy|\sqrt{x^2+y^2}+|yz|\sqrt{y^2+z^2}+|zx|\sqrt{z^2+x^2}.\] Find the best constants $c_1,c_2\in\mathbb{R}$ such that \[c_1(x^2+y^2+z^2)^{3/2}\leq f(x,y,z)\leq c_1(x^2+y^2+z^2)^{3/2}\] hold for all reals $x,y,z$ satisfying $x+y+z=0$. [i]Proposed by Untro368.[/i]

2024 Greece Junior Math Olympiad, 1

Tags: algebra , inequality

a) Prove that for all real numbers $k,l,m$ holds : $$(k+l+m)^2 \ge 3 (kl+lm+mk)$$ When does equality holds? b) If $x,y,z$ are positive real numbers and $a,b$ real numbers such that $$a(x+y+z)=b(xy+yz+zx)=xyz,$$ prove that $a \ge 3b^2$. When does equality holds?

2021 Israel TST, 1

Tags: inequalities , algebra

Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\] or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]

2000 District Olympiad (Hunedoara), 1

Tags: inequalities , inequality , algebra

[b]a)[/b] Show that $ \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, $ for all non-negative reals $ x $ and integers $ n\ge 2. $ [b]b)[/b] If $ x,y,z\in (0,\infty ) , $ then prove the inequality $$ \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16 $$