Found problems: 110
2014 IMC, 4
We say that a subset of $\mathbb{R}^n$ is $k$-[i]almost contained[/i] by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-[i]generic[/i] if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $(k, n)$, find the minimal number of $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^n$ contains a $k$-generic subset with at most $d(k, n)$ elements.
(Proposed by Shachar Carmeli, Weizmann Inst. and Lev Radzivilovsky, Tel Aviv Univ.)
2001 Miklós Schweitzer, 1
Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$, then $f$ assumes at most $|S|$ distinct values.
2005 Bosnia and Herzegovina Team Selection Test, 5
If for an arbitrary permutation $(a_1,a_2,...,a_n)$ of set ${1,2,...,n}$ holds $\frac{{a_k}^2}{a_{k+1}}\leq k+2$,
$k=1,2,...,n-1$, prove that $a_k=k$ for $k=1,2,...,n$
2020 Bangladesh Mathematical Olympiad National, Problem 9
Bristy wants to build a special set $A$. She starts with $A=\{0, 42\}$. At any step, she can add an integer $x$ to the set $A$ if it is a root of a polynomial which uses the already existing integers in $A$ as coefficients. She keeps doing this, adding more and more numbers to $A$. After she eventually runs out of numbers to add to $A$, how many numbers will be in $A$?
1989 Balkan MO, 4
The elements of the set $F$ are some subsets of $\left\{1,2,\ldots ,n\right\}$ and satisfy the conditions:
i) if $A$ belongs to $F$, then $A$ has three elements;
ii)if $A$ and $B$ are distinct elements of $F$ , then $A$ and $B$ have at most one common element.
Let $f(n)$ be the greatest possible number of elements of $F$. Prove that $\frac{n^{2}-4n}{6}\leq f(n) \leq \frac{n^{2}-n}{6}$
2010 IMC, 4
Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.
2020 Taiwan APMO Preliminary, P5
Let $S$ is the set of permutation of {1,2,3,4,5,6,7,8}
(1)For all $\sigma=\sigma_1\sigma_2...\sigma_8\in S$
Evaluate the sum of S=$\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8$. Then for all elements in $S$,what is the arithmetic mean of S?
(Notice $S$ and S are different.)
(2)In $S$, how many permutations are there which satisfies "For all $k=1,2,...,7$,the digit after k is [b]not[/b] (k+1)"?
1986 Traian Lălescu, 1.4
Let be a parametric set:
$$ \mathcal{F}_{\lambda } =\left\{ f:[1,\infty)\longrightarrow\mathbb{R}\bigg| x\in(1,\infty )\implies \int_{x}^{x^2+\lambda^2 x} f\left( \xi\right) d\xi =1\right\} . $$
[b]a)[/b] Show that $ \mathcal{F}_0 =\emptyset . $
[b]b)[/b] Prove that $ \lambda\neq 0 $ implies $ \mathcal{F}_{\lambda }\neq\emptyset . $
2015 Greece JBMO TST, 4
Pupils of a school are divided into $112$ groups, of $11$ members each.
Any two groups have exactly one common pupil. Prove that:
a) there is a pupil that belongs to at least $12$ groups.
b) there is a pupil that belongs to all the groups.
2002 District Olympiad, 4
For any natural number $ n\ge 2, $ define $ m(n) $ to be the minimum number of elements of a set $ S $ that simultaneously satisfy:
$ \text{(i)}\quad \{ 1,n\} \subset S\subset \{ 1,2,\ldots ,n\} $
$ \text{(ii)}\quad $ any element of $ S, $ distinct from $ 1, $ is equal to the sum of two (not necessarily distinct) elements from $ S. $
[b]a)[/b] Prove that $ m(n)\ge 1+\left\lfloor \log_2 n \right\rfloor ,\quad\forall n\in\mathbb{N}_{\ge 2} . $
[b]b)[/b] Prove that there are infinitely many natural numbers $ n\ge 2 $ such that $ m(n)=m(n+1). $
$ \lfloor\rfloor $ denotes the usual integer part.