Found problems: 15925
2013 ELMO Shortlist, 8
We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets.
For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$.
[i]Proposed by Victor Wang[/i]
1989 Irish Math Olympiad, 3
A function $f$ is defined on the natural numbers $\mathbb{N}$ and satisfies the following rules:
(a) $f(1)=1$;
(b) $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$ for all $n\in \mathbb{N}$.
Calculate the maximum value $m$ of the set $\{f(n):n\in \mathbb{N}, 1\le n\le 1989\}$, and determine the number of natural numbers $n$, with $1\le n\le 1989$, that satisfy the equation $f(n)=m$.
2011 Middle European Mathematical Olympiad, 8
We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality
\[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\]
holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i].
[b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.
2023 VN Math Olympiad For High School Students, Problem 3
Given a polynomial with integer coefficents with degree $n>0:$$$P(x)=a_nx^n+...+a_1x+a_0.$$
Assume that there exists a prime number $p$ satisfying these conditions:
[i]i)[/i] $p|a_i$ for all $0\le i<n,$
[i]ii)[/i] $p\nmid a_n,$
[i]iii)[/i] $p^2\nmid a_0.$
Prove that $P(x)$ is irreducible in $\mathbb{Z}[x].$
2019 Moldova Team Selection Test, 9
Find all polynomials $P(X)$ with real coefficients such that if real numbers $x,y$ and $z$ satisfy $x+y+z=0,$ then the points $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$ are all colinear.
2003 Estonia Team Selection Test, 3
Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows:
(i) $A(0, m, n) = m'$ for all $m, n \in N$
(ii) $A(k', 0, n) =\left\{ \begin{array}{ll}
n & if \, \, k = 0 \\
0 & if \, \,k = 1, \\
1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$
(iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$.
Compute $A(5, 3, 2)$.
(H. Nestra)
1995 Belarus Team Selection Test, 3
If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: \[a^pb^q+(1-a)^p(1-b)^q\le 1\]
2019 District Olympiad, 1
Find the functions $f: \mathbb{R} \to (0, \infty)$ which satisfy $$2^{-x-y} \le \frac{f(x)f(y)}{(x^2+1)(y^2+1)} \le \frac{f(x+y)}{(x+y)^2+1},$$ for all $x,y \in \mathbb{R}.$
2021 Korea Winter Program Practice Test, 2
Find all functions $f:R^+\rightarrow R^+$ such that for all positive reals $x$ and $y$
$$4f(x+yf(x))=f(x)f(2y)$$
2022 BMT, 19-21
[center][u]Guts Round[/u] / [u]Set 7[/u][/center]
[b]p19.[/b] Let $N \ge 3$ be the answer to Problem 21.
A regular $N$-gon is inscribed in a circle of radius $1$. Let $D$ be the set of diagonals, where we include all sides as diagonals. Then, let $D'$ be the set of all unordered pairs of distinct diagonals in $D$. Compute the sum $$\sum_{\{d,d'\}\in D'} \ell (d)^2 \ell (d')^2,$$where $\ell (d)$ denotes the length of diagonal $d$.
[b]p20.[/b] Let $N$ be the answer to Problem $19$, and let $M$ be the last digit of $N$.
Let $\omega$ be a primitive $M$th root of unity, and define $P(x)$ such that$$P(x) = \prod^M_{k=1} (x - \omega^{i_k}),$$where the $i_k$ are chosen independently and uniformly at random from the range $\{0, 1, . . . ,M-1\}$. Compute $E \left[P\left(\sqrt{\rfloor \frac{1250}{N} \rfloor } \right)\right].$
[b]p21.[/b] Let $N$ be the answer to Problem $20$.
Define the polynomial $f(x) = x^{34} +x^{33} +x^{32} +...+x+1$. Compute the number of primes $p < N$ such that there exists an integer $k$ with $f(k)$ divisible by $p$.
2016 Estonia Team Selection Test, 6
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.
2012 Polish MO Finals, 6
Show that for any positive real numbers $a, b, c$ true is inequality:
$\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$.
2009 Ukraine National Mathematical Olympiad, 2
Find all functions $f : \mathbb Z \to \mathbb Z$ such that
\[f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.\]
[b]Note.[/b] $|x|$ denotes the absolute value of the integer $x.$
2021/2022 Tournament of Towns, P5
What is the maximal possible number of roots on the interval (0,1) for a polynomial of degree 2022 with integer coefficients and with the leading coefficient equal to 1?
2013 Saudi Arabia BMO TST, 8
Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.
2023 AMC 10, 5
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$. Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$. How many numbers are written on the blackboard?
$\textbf{(A) }10\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2021 Czech and Slovak Olympiad III A, 3
The different non-zero real numbers a, b, c satisfy the set equality $\{a + b, b + c, c + a\} = \{ab, bc, ca\}$.
Prove that the set equality $\{a, b, c\} = \{a^2 -2, b^2 - 2, c^2 - 2\}$ also holds.
.
(Josef Tkadlec)
2010 Tuymaada Olympiad, 4
Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.
2021-IMOC qualification, A3
Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$
2010 Saudi Arabia IMO TST, 1
Find all real numbers $x$ that can be written as $$x= \frac{a_0}{a_1a_2..a_n}+\frac{a_1}{a_2a_3..a_n}+\frac{a_2}{a_3a_4..a_n}+...+\frac{a_{n-2}}{a_{n-1}a_n}+\frac{a_{n-1}}{a_n}$$
where $n, a_1,a_2,...,a_n$ are positive integers and $1 = a_0 \le a_1 <... < a_n$
2016 Regional Olympiad of Mexico West, 2
Let $A$ be an infinite set of real numbers containing at least one irrational number. Prove that for every natural number $n > 1$ there exists a subset $S$ of $A$ with n elements such that the sum of the elements of $S$ is an irrational number.
2011 AIME Problems, 15
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.
2020 Bulgaria Team Selection Test, 2
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$
2006 Lithuania National Olympiad, 3
Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.
2018 India IMO Training Camp, 3
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.