Found problems: 126
2009 Iran MO (2nd Round), 3
$11$ people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and $11$ cards with numbers $1$ to $11$ are given to them. Some may have no card and some may have more than $1$ card. In each round, one [and only one] can give one of his cards with number $ i $ to his adjacent person if after and before the round, the locations of the cards with numbers $ i-1,i,i+1 $ don’t make an acute-angled triangle.
(Card with number $0$ means the card with number $11$ and card with number $12$ means the card with number $1$!)
Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.)
Prove that the cards can’t be gathered at one person.
1964 Miklós Schweitzer, 9
Let $ E$ be the set of all real functions on $ I\equal{}[0,1]$. Prove that one cannot define a topology on $ E$ in which $ f_n\rightarrow f$ holds if and only if $ f_n$ converges to $ f$ almost everywhere.
2024 Miklos Schweitzer, 5
Let $X$ be a regular topological space and let $S$ be a countably compact dense subspace in $X$. (The countably compact property means that every infinite subset of $S$ has an accumulation point in $S$.) Show that $S$ is also $G_\delta$-dense in $X$, i.e., $S$ intersects all nonempty $G_\delta$ sets.
2005 Iran MO (3rd Round), 1
We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$.
a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN.
b) The circle is not CN.
Which one of these sets are CN?
1) $A=\{x\in\mathbb R^3| |x|=1\}$
2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$
3) Graph of the function $f:[0,1]\to \mathbb R$ defined by
\[f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0\]
1964 Putnam, A3
Let $P_1 , P_2 , \ldots$ be a sequence of distinct points which is dense in the interval $(0,1)$. The points $P_1 , \ldots , P_{n-1}$ decompose the interval into $n$ parts, and $P_n$ decomposes one of these into two parts. Let $a_n$ and $b_n$ be the length of these two intervals. Prove that
$$\sum_{n=1}^{\infty} a_n b_n (a_n +b_n) =1 \slash 3.$$
2008 IMS, 9
Let $ \gamma: [0,1]\rightarrow [0,1]\times [0,1]$ be a mapping such that for each $ s,t\in [0,1]$
\[ |\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha
\]
in which $ \alpha,M$ are fixed numbers. Prove that if $ \gamma$ is surjective, then $ \alpha\leq\frac12$
2010 Miklós Schweitzer, 10
Consider the space $ \{0,1 \} ^{N} $ with the product topology (where $\{0,1 \}$ is a discrete space). Let $ T: \{0,1 \} ^ {\mathbb {N}} \rightarrow \{0,1 \} ^ {\mathbb {N}} $ be the left-shift, ie $ (Tx) (n) = x (n+1) $ for every $ n \in \mathbb {N} $.
Can a finite number of Borel sets be given: $ B_ {1}, \ldots, B_ {m} \subset \{0,1 \} ^ {N} $ such that
$$
\left \{T ^ {i} \left (B_ {j} \right) \mid i \in \mathbb {N}, 1 \leq j \leq m \right \}
$$the $ \sigma $-algebra generated by the set system coincides with the Borel set system?
2012 Putnam, 3
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that
(i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$
(ii) $ f(0)=1,$ and
(iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
Prove that $f$ is unique, and express $f(x)$ in closed form.
2009 Miklós Schweitzer, 10
Let $ U\subset\mathbb R^n$ be an open set, and let $ L: U\times\mathbb R^n\to\mathbb R$ be a continuous, in its second variable first order positive homogeneous, positive over $ U\times (\mathbb R^n\setminus\{0\})$ and of $ C^2$-class Langrange function, such that for all $ p\in U$ the Gauss-curvature of the hyper surface
\[ \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \}\]
is nowhere zero. Determine the extremals of $ L$ if it satisfies the following system
\[ \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\})\]
of partial differetial equations, where $ y^k(u,v) : \equal{} v^k$ for $ (u,v)\in U\times\mathbb R^k$, $ v \equal{} (v^1,\dots,v^k)$.
2005 VJIMC, Problem 1
Let $S_0=\{z\in\mathbb C:|z|=1,z\ne-1\}$ and $f(z)=\frac{\operatorname{Im}z}{1+\operatorname{Re}z}$. Prove that $f$ is a bijection between $S_0$ and $\mathbb R$. Find $f^{-1}$.
1982 Miklós Schweitzer, 1
A map $ F : P(X) \rightarrow P(X)$, where $ P(X)$ denotes the set of all subsets of $ X$, is called a $ \textit{closure operation}$ on $ X$ if for arbitrary $ A,B \subset X$, the following conditions hold:
(i) $ A \subset F(A);$
(ii) $ A \subset B \Rightarrow F(A) \subset F(B);$
(iii) $ F(F(A))\equal{}F(A)$.
The cardinal number $ \min \{ |A| : \;A \subset X\ ,\;F(A)\equal{}X\ \}$ is called the $ \textit{density}$ of $ F$ and is denoted by $ d(F)$. A set $ H \subset X$ is called $ \textit{discrete}$ with respect to $ F$ if $ u \not \in F(H\minus{}\{ u \})$ holds for all $ u \in H$. Prove that if the density of the closure operation $ F$ is a singular cardinal number, then for any nonnegative integer $ n$, there exists a set of size $ n$ that is discrete with respect to $ F$. Show that the statement is not true when the existence of an infinite discrete subset is required, even if $ F$ is the closure operation of a topological space satisfying the $ T_1$ separation axiom.
[i]A. Hajnal[/i]
2008 Miklós Schweitzer, 8
Let $S$ be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets $f^{-1}(y)$ for typical continuous real functions defined on $S$? (A property is satisfied for typical continuous real functions on $S$ if the set of functions not having this property is of the first Baire category in the metric space of continuous $S\rightarrow\mathbb{R}$ functions with the supremum norm.)
(translated by Miklós Maróti)
1964 Putnam, B3
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?
2011 Pre-Preparation Course Examination, 1
[b]a)[/b] prove that for every compressed set $K$ in the space $\mathbb R^3$, the function $f:\mathbb R^3 \longrightarrow \mathbb R$ that $f(p)=inf\{|p-k|,k\in K\}$ is continuous.
[b]b)[/b] prove that we cannot cover the sphere $S^2\subseteq \mathbb R^3$ with it's three closed sets, such that none of them contain two antipodal points.
MIPT student olimpiad spring 2023, 4
Is it true that if two linear subspaces $V$ and $W$ of a Hilbert space are closed, then their sum $V+W$ is also closed?
2010 Miklós Schweitzer, 8
Let $ D \subset \mathbb {R} ^ {2} $ be a finite Lebesgue measure of a connected open set and $ u: D \rightarrow \mathbb {R} $ a harmonic function. Show that it is either a constant $ u $ or for almost every $ p \in D $
$$
f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), \quad f (0) = p
$$has no initial value problem(differentiable everywhere) solution to $ f:[0,\infty) \rightarrow D $.
MIPT student olimpiad autumn 2022, 1
Prove that if a function $f:R \to R$ is bounded and its graph is closed as
subset of the $R^2$ plane, then the function f is continuous.
2011 Pre-Preparation Course Examination, 2
prove that $\pi_1 (X,x_0)$ is not abelian. $X$ is like an eight $(8)$ figure.
[b]comments:[/b] eight figure is the union of two circles that have one point $x_0$ in common.
we call a group $G$ abelian if: $\forall a,b \in G:ab=ba$.
2012 India National Olympiad, 1
Let $ABCD$ be a quadrilateral inscribed in a circle. Suppose $AB=\sqrt{2+\sqrt{2}}$ and $AB$ subtends $135$ degrees at center of circle . Find the maximum possible area of $ABCD$.
1982 IMO Shortlist, 19
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$
2008 ISI B.Stat Entrance Exam, 10
Two subsets $A$ and $B$ of the $(x,y)$-plane are said to be [i]equivalent[/i] if there exists a function $f: A\to B$ which is both one-to-one and onto.
(i) Show that any two line segments in the plane are equivalent.
(ii) Show that any two circles in the plane are equivalent.
1981 Miklós Schweitzer, 8
Let $ W$ be a dense, open subset of the real line $ \mathbb{R}$. Show that the following two statements are equivalent:
(1) Every function $ f : \mathbb{R} \rightarrow \mathbb{R}$ continuous at all points of $ \mathbb{R} \setminus W$ and nondecreasing on every open interval contained in $ W$ is nondecreasing on the whole $ \mathbb{R}$.
(2) $ \mathbb{R} \setminus W$ is countable.
[i]E. Gesztelyi[/i]
2014 ELMO Shortlist, 1
In a non-obtuse triangle $ABC$, prove that
\[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]
2003 SNSB Admission, 4
Prove that the sets
$$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2+x_3^2=x_4^2 \} , $$
$$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2=x_3^2+x_4^2 \}, $$
are not homeomorphic on the Euclidean topology induced on them.
2009 Spain Mathematical Olympiad, 3
Some edges are painted in red. We say that a coloring of this kind is [i]good[/i], if for each vertex of the polyhedron, there exists an edge which concurs in that vertex and is not painted red. Moreover, we say that a coloring where some of the edges of a regular polyhedron is [i]completely good[/i], if in addition to being [i]good[/i], no face of the polyhedron has all its edges painted red. What regular polyhedrons is equal the maximum number of edges that can be painted in a [i]good[/i] color and a [i]completely good[/i]? Explain your answer.