Olympiad problems

2003 Estonia National Olympiad, 4

Tags: integer , divide , infinity , divisible , number theory , floor function

Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.

1997 Miklós Schweitzer, 6

Tags: infinity , function

Let $\kappa$ be an infinite cardinality and let A , B be sets of cardinality $\kappa$. Construct a family $\cal F$ of functions $f : A \to B$ with cardinality $2^\kappa$ such that for all functions $f_1,\cdots, f_n \in\cal F$ and for all $b_1 , ..., b_n \in B$, there exist $a\in A$ such that $f_1(a) = b_1,\cdots, f_n(a) = b_n$.

1999 VJIMC, Problem 3

Tags: infinity , geometry

Suppose that we have a countable set $A$ of balls and a unit cube in $\mathbb R^3$. Assume that for every finite subset $B$ of $A$ it is possible to put all balls of $B$ into the cube in such a way that they have disjoint interiors. Show that it is possible to arrange all the balls in the cube so that all of them have pairwise disjoint interiors.

2018 Regional Olympiad of Mexico Southeast, 4

Tags: infinity , number theory

For every natural $n$ let $a_n=20\dots 018$ with $n$ ceros, for example, $a_1=2018, a_3=200018, a_7=2000000018$. Prove that there are infinity values of $n$ such that $2018$ divides $a_n$

2022 Indonesia TST, N

Tags: number theory , infinity , product , prime , polynomial

Let $n$ be a natural number, with the prime factorisation \[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define \[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.

1989 Putnam, B4

Tags: infinity , set theory

Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?

2000 VJIMC, Problem 1

Tags: real analysis , set , set theory , infinity

Is there a countable set $Y$ and an uncountable family $\mathcal F$ of its subsets such that for every two distinct $A,B\in\mathcal F$, their intersection $A\cap B$ is finite?

1959 Putnam, B3

Tags: function , continuity , infinity

Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.

2010 German National Olympiad, 3

Tags: combinatorics , set , infinite set , book , infinity

An infinite fairytale is a book with pages numbered $1,2,3,\ldots$ where all natural numbers appear. An author wants to write an infinite fairytale such that a new dwarf is introduced on each page. Afterward, the page contains several discussions between groups of at least two of the already introduced dwarfs. The publisher wants to make the book more exciting and thus requests the following condition: Every infinite set of dwarfs contains a group of at least two dwarfs, who formed a discussion group at some point as well as a group of the same size for which this is not true. Can the author fulfill this condition?