Olympiad problems

2004 Finnish National High School Mathematics Competition, 1

Tags: quadratic , algebra

The equations $x^2 +2ax+b^2 = 0$ and $x^2 +2bx+c^2 = 0$ both have two different real roots. Determine the number of real roots of the equation $x^2 + 2cx + a^2 = 0.$

2022 BMT, 14

Tags: algebra

Isaac writes each fraction $\frac{1^2}{300}$ , $\frac{2^2}{300}$ , $...$, $\frac{300^2}{300}$ in reduced form. Compute the sum of all denominators over all the reduced fractions that Isaac writes down.

2010 IFYM, Sozopol, 8

Tags: polynomial , algebra

Find all polynomials $f(x)$ with integer coefficients and leading coefficient equal to 1, for which $f(0)=2010$ and for each irrational $x$, $f(x)$ is also irrational.

2000 German National Olympiad, 4

Tags: algebra , system of equations

Find all nonnegative solutions $(x,y,z)$ to the system $$\begin{cases} \sqrt{x+y}+\sqrt{z} = 7 \\ \sqrt{x+z}+\sqrt{y} = 7 \\ \sqrt{y+z}+\sqrt{x} = 5 \end{cases}$$

2024 Iran MO (3rd Round), 1

Tags: functional equation , complex number , fe , function , algebra

Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and: $$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$ Where $\overline{a}$ is the complex conjugate of $a$.

2013 Estonia Team Selection Test, 3

Tags: inequalities , algebra , sum

Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros. (i) Prove that $$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$ (ii) Show that, for each $n > 1$, both inequalities can hold as equalities.

2023 Malaysian IMO Training Camp, 2

Tags: algebra

Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$) [i]Proposed by Wong Jer Ren[/i]

2023 Korea - Final Round, 6

Tags: inequalities , algebra

For positive integer $n\geq 3$ and real numbers $a_1,...,a_n,b_1,...,b_n$, prove the following. $$\sum_{i=1}^n a_i(b_i-b_{i+3})\leq\frac{3n}{8}\sum_{i=1}^n((a_i-a_{i+1})^2+(b_i-b_{i+1})^2)$$ ($a_{n+1}=a_1$, and for $i=1,2,3$ $b_{n+i}=b_i$.)

2016 Postal Coaching, 4

Tags: function , functional equation , algebra

Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.

2018 JBMO Shortlist, NT1

Tags: number theory , algebra

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

2005 MOP Homework, 2

Tags: floor function , algebra , inequalities

Find all real numbers $x$ such that $\lfloor x^2-2x \rfloor+2\lfloor x \rfloor=\lfloor x \rfloor^2$. (For a real number $x$, $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$.)

1978 IMO Longlists, 47

Tags: algebra , polynomial , induction

Given the expression \[P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],\] prove: $(a) P_n(x)$ satisfies the identity \[P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.\] $(b) P_n(x)$ is a polynomial in $x$ of degree $n.$

2012 Belarus Team Selection Test, 3

Tags: trigonometry , triangle inequality , triangle , algebra

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

2004 Tournament Of Towns, 3

Tags: polynomial , algebra

P(x) and Q(x) are polynomials of positive degree such that for all x P(P(x))=Q(Q(x)) and P(P(P(x)))=Q(Q(Q(x))). Does this necessarily mean that P(x)=Q(x)?

1988 IMO Longlists, 10

Tags: polynomial , number theory , divisibility , algebra , modular arithmetic

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

1971 Poland - Second Round, 2

Tags: algebra , trigonometry , inequalities

Prove that if $ A, B, C $ are angles of a triangle, then $$ 1 < \cos A + \cos B + \cos C \leq \frac{3}{2}.$$

2010 Germany Team Selection Test, 3

Tags: number theory , polynomial , permutation , algebra

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

2010 IMO Shortlist, 5

Tags: number theory , perfect square , function , algebra

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

2012 Iran Team Selection Test, 2

Tags: induction , binomial coefficient , algebra , ceiling function , function

The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$: [b]a)[/b] $f(a)=0 \Leftrightarrow a=0$ [b]b)[/b] $f(ab)=f(a)f(b)$ [b]c)[/b] $f(a+b)\le 2 \max \{f(a),f(b)\}$. Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$. [i]Proposed by Masoud Shafaei[/i]

1990 IberoAmerican, 6

Tags: polynomial , algebra

Let $f(x)$ be a cubic polynomial with rational coefficients. If the graph of $f(x)$ is tangent to the $x$ axis, prove that the roots of $f(x)$ are all rational.

1981 Swedish Mathematical Competition, 2

Tags: system of equations , system , algebra

Does \[\left\{ \begin{array}{l} x^y = z \\ y^z = x \\ z^x = y \\ \end{array} \right. \] have any solutions in positive reals apart from $x = y = z= 1$?

2005 Estonia Team Selection Test, 4

Tags: algebra , polynomial , real root

Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.

Kvant 2019, M2544

Tags: inequalities , polynomial , algebra

Let $P(x)=x^n +a_1x^{n-1}+a_2x^{n-2}+\ldots+a_{n-1}x+a_n$ be a polynomial of degree $n$ and $n$ real roots, all of them in the interval $(0,1)$. Prove that for all $k=\overline{1,n}$ the following inequality holds: \[(-1)^k(a_k+a_{k+1}+\ldots+a_n)>0.\] [i]Proposed by N. Safaei (Iran)[/i]

1997 Austrian-Polish Competition, 7

Tags: inequalities , max , algebra

(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, . (b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$ (c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.

2021 Peru IMO TST, P1

Tags: inequalities , algebra

Suppose positive real numers $x,y,z,w$ satisfy $(x^3+y^3)^4=z^3+w^3$. Prove that $$x^4z+y^4w\geq zw.$$