Found problems: 15925
2019 Estonia Team Selection Test, 4
Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.
2007 IMC, 2
Let $ x$, $ y$ and $ z$ be integers such that $ S = x^{4}+y^{4}+z^{4}$ is divisible by 29. Show that $ S$ is divisible by $ 29^{4}$.
2005 ITAMO, 2
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and
\[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\]
For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?
1996 Romania National Olympiad, 2
Find all real numbers $x$ for which the following equality holds :
$$\sqrt{\frac{x-7}{1989}}+\sqrt{\frac{x-6}{1990}}+\sqrt{\frac{x-5}{1991}}=\sqrt{\frac{x-1989}{7}}+\sqrt{\frac{x-1990}{6}}+\sqrt{\frac{x-1991}{5}}$$
2021 Alibaba Global Math Competition, 16
Let $G$ be a finite group, and let $H_1, H_2 \subset G$ be two subgroups. Suppose that for any representation of $G$ on a finite-dimensional complex vector space $V$, one has that
\[\text{dim} V^{H_1}=\text{dim} V^{H_2},\]
where $V^{H_i}$ is the subspace of $H_i$-invariant vectors in $V$ ($i=1,2$). Prove that
\[Z(G) \cap H_1=Z(G) \cap H_2.\]
Here $Z(G)$ denotes the center of $G$.
2004 Federal Competition For Advanced Students, Part 1, 4
Each of the $2N = 2004$ real numbers $x_1, x_2, \ldots , x_{2004}$ equals either $\sqrt 2 -1 $ or $\sqrt 2 +1$. Can the sum $\sum_{k=1}^N x_{2k-1}x_2k$ take the value $2004$? Which integral values can this sum take?
1964 IMO Shortlist, 2
Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]
2010 Benelux, 2
Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]
2019 Harvard-MIT Mathematics Tournament, 8
There is a unique function $f: \mathbb{N} \to \mathbb{R}$ such that $f(1) > 0$ and such that
\[\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right) = 1\]
for all $n \ge 1$. What is $f(2018^{2019})$?
2020 South East Mathematical Olympiad, 3
Given a polynomial $f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i$, where $c_i \in \{ -1,0,1 \}$. Denote $N$ the number of positive integer roots of $f(x)=0$ (counting multiplicity). If $f(x)=0$ has no negative integer roots, find the maximum of $N$.
2012 Mathcenter Contest + Longlist, 5 sl13
Define $f : \mathbb{R}^+ \rightarrow \mathbb{R}$ as the strictly increasing function such that
$$f(\sqrt{xy})=\frac{f(x)+f(y)}{2}$$ for all positive real numbers $x,y$. Prove that there are some positive real numbers $a$ where $f(a)<0$.
[i] (PP-nine) [/i]
2024 Romania EGMO TST, P1
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i]Proposed by Nikolai Nikolov, Bulgaria[/i]
2016 Iran MO (3rd Round), 3
Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb {R}^{+} $ such that for all positive real numbers $x,y:$
$$f(y)f(x+f(y))=f(x)f(xy)$$
2024 Rioplatense Mathematical Olympiad, 3
Given a set $S$ of integers, an allowed operation consists of the following three steps:
$\bullet$ Choose a positive integer $n$.
$\bullet$ Choose $n+1$ elements $a_0, a_1, \dots, a_n \in S$, not necessarily distinct.
$\bullet$ Add to the set $S$ all the integer roots of the polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$.
Beto must choose an initial set $S$ and perform several allowed operations, so that at the end of the process $S$ contains among its elements the integers $1, 2, 3, \dots, 2023, 2024$.
Determine the smallest $k$ for which there exists an initial set $S$ with $k$ elements that allows Beto to achieve his objective.
2023 Thailand Mathematical Olympiad, 3
Defined all $f : \mathbb{R} \to \mathbb{R} $ that satisfied equation $$f(x)f(y)f(x-y)=x^2f(y)-y^2f(x)$$ for all $x,y \in \mathbb{R}$
2013 Romania Team Selection Test, 4
Let $k$ be a positive integer larger than $1$. Build an infinite set $\mathcal{A}$ of subsets of $\mathbb{N}$ having the following properties:
[b](a)[/b] any $k$ distinct sets of $\mathcal{A}$ have exactly one common element;
[b](b)[/b] any $k+1$ distinct sets of $\mathcal{A}$ have void intersection.
1978 All Soviet Union Mathematical Olympiad, 268
Consider a sequence $$x_n=(1+\sqrt2+\sqrt3)^n$$ Each member can be represented as $$x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$$ where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.
2019 India PRMO, 15
In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ?
2017 Baltic Way, 1
Let $a_0,a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2}\geq a_n$ for all positive integers $n$. Show that $$\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}$$ holds for all positive integers $n$.
2017 Irish Math Olympiad, 5
Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies
$$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$.
(a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero.
(b) Find a sequence none of whose terms are zero but which is $2017$-powerful.
1997 Bosnia and Herzegovina Team Selection Test, 4
$a)$ In triangle $ABC$ let $A_1$, $B_1$ and $C_1$ be touching points of incircle $ABC$ with $BA$, $CA$ and $AB$, respectively. Let $l_1$, $l_2$ and $l_3$ be lenghts of arcs $ B_1C_1$, $A_1C_1$, $B_1A_1$ of incircle $ABC$, respectively, which does not contain points $A_1$, $B_1$ and $C_1$, respectively.
Does the following inequality hold: $$ \frac{a}{l_1}+\frac{b}{l_2}+\frac{c}{l_3} \geq \frac{9\sqrt{3}}{\pi}$$
$b)$ Tetrahedron $ABCD$ has three pairs of equal opposing sides. Find length of height of tetrahedron in function od lengths of sides
1955 Moscow Mathematical Olympiad, 307
* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.
2006 Victor Vâlcovici, 1
Prove that for any real numbers $ a,b,c, $ the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\sqrt{(x-c)^2+b^2} +\sqrt{(x+c)^2+b^2} $$ is decreasing on $ (-\infty ,0] $ and increasing on $ [0,\infty ) . $
2007 Mathematics for Its Sake, 2
Let be a natural number $ k $ and let be two infinite sequences $ \left( x_n \right)_{n\ge 1} ,\left( y_n \right)_{n\ge 1} $ such that
$$ \{1\}\cap\{ x_1,x_2,\ldots ,x_k\}=\{1\}\cap\{ y_1,y_2,\ldots ,y_k\} =\{ x_1,x_2,\ldots ,x_k\}\cap\{ y_1,y_2,\ldots ,y_k\} =\emptyset , $$
and defined by the following recurrence relations:
$$ x_{n+k}=\frac{y_n}{x_n} ,\quad y_{n+k} =\frac{y_n-1}{x_n-1} $$
Prove that $ \left( x_n \right)_{n\ge 1} $ and $ \left( y_n \right)_{n\ge 1} $ are periodic.
[i]Dumitru Acu[/i]
1990 Baltic Way, 11
Prove that the modulus of an integer root of a polynomial with integer coefficients cannot exceed the maximum of the moduli of the coefficients.