Found problems: 876
1953 Miklós Schweitzer, 7
[b]7.[/b] Consider four real numbers $t_{1},t_{2},t_{3},t_{4}$ such that each is less than the sum of the others. Show that there exists a tetrahedron whose faces have areas $t_{1},t_{2}, t_{3}$ and $t_{4},$ respectively. [b](G. 9)[/b]
1958 Miklós Schweitzer, 6
[b]6.[/b] Prove that if $a_n \geq 0$ and
$\frac{1}{n}\sum_{k=1}^{n} a_k \geq \sum_{k=n+1}^{2n}a_k$ $(n=1, 2, \dots)$ ,
then $\sum_{k=1}^{\infty} a_k $ is convergent and its sum is less than $2ea_1$. [b](S. 9)[/b]
1982 Putnam, A6
Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties:
$(\text i)$ $|x_n|$ is a strictly decreasing function of $n$;
$(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$;
$(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$.
Prove or disprove that these conditions imply that
$$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$
2012 Putnam, 3
A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?
1966 Putnam, B3
Show that if the series $$\sum_{n=1}^{\infty} \frac{1}{p_n}$$ is convergent, where $p_1,p_2,p_3,\dots, p_n, \dots$ are positive real numbers, then the series $$\sum_{n=1}^{\infty} \frac{n^2}{(p_1+p_2+\dots +p_n)^2}p_n$$ is also convergent.
2021 Alibaba Global Math Competition, 10
In $\mathbb{R}^3$, for a rectangular box $\Delta$, let $10\Delta$ be the box with the same center as $\Delta$ but dilated by $10$. For example, if $\Delta$ is an $1 \times 1 \times 10$ box (hence with Lebesgue measure $10$), then $10\Delta$ is the $10 \times 10 \times 100$ box with the same center and orientation as $\Delta$.
\medskip
If two rectangular boxes $\Delta_1$ and $\Delta_2$ satisfy $\Delta_1 \subset 10\Delta_2$ and $\Delta_2 \subset 10 \Delta_1$, we say that they are [i]almost identical[/i].
Find the largest real number $a$ such that the following holds for some $C=C(a)>0$:
For every positive integer $N$ and every collection $S$ of $1 \times 1 \times N$ boxes in $\mathbb{R}^3$, assuming that (i) $\vert S\vert=N$, (ii) every pair of boxes $(\Delta_1,\Delta_2)$ taken from $S$ are not almost identical, and (iii) the long edge of each box in $S$ forms an angle $\frac{\pi}{4}$ against the $xy$-plane. Then the volume
\[\left\vert \bigcup_{\Delta \in S} \Delta\right\vert \ge CN^a.\]
1994 IMC, 3
Given a set $S$ of $2n-1$, $n\in \mathbb N$, different irrational numbers. Prove that there are $n$ different elements $x_1, x_2, \ldots, x_n\in S$ such that for all non-negative rational numbers $a_1, a_2, \ldots, a_n$ with $a_1+a_2+\ldots + a_n>0$ we have that $a_1x_1+a_2x_2+\cdots +a_nx_n$ is an irrational number.
2022 District Olympiad, P2
Let $(G,\cdot)$ be a group and $H\neq G$ be a subgroup so that $x^2=y^2$ for all $x,y\in G\setminus H.$ Show that $(H,\cdot)$ is an Abelian group.
1994 IMC, 6
Find
$$\lim_{N\to\infty}\frac{\ln^2 N}{N} \sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)}$$
MIPT student olimpiad spring 2024, 3
Is it true that if a function $f: R \to R$ is continuous and takes rational values at rational points,
then at least at one point it is differentiable?
ICMC 3, 2
Let \(\mathbb{R}^2\) denote the set of points in the Euclidean plane. For points \(A,P\in\mathbb{R}^2\) and a real number \(k\), define the [i]dilation[/i] of \(A\) about \(P\) by a factor of \(k\) as the point \(P+k(A-P)\). Call a sequence of point \(A_0, A_1, A_2,\ldots\in\mathbb{R}^2\) [i]unbounded[/i] if the sequence of lengths \(\left|A_0-A_0\right|,\left|A_1-A_0\right|,\left|A_2-A_0\right|,\ldots\) has no upper bound.
Now consider \(n\) distinct points \(P_0,P_1,\ldots,P_{n-1}\in\mathbb{R}^2\), and fix a real number \(r\). Given a starting point \(A_0\in\mathbb{R}^2\), iteratively define \(A_{i+1}\) by dilating \(A_i\) about \(P_j\) by a factor of \(r\), where \(j\) is the remainder of \(i\) when divided by \(n\).
Prove that if \(\left|r\right|\geq 1\), then for any starting point \(A_0\in\mathbb{R}^2\), the sequence \(A_0,A_1,A_2,\ldots\) is either periodic or unbounded.
[i]Proposed by the ICMC Problem Committee[/i]
2005 Putnam, B1
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$
(Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)
1980 Miklós Schweitzer, 1
For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there exist infinitely many pairs $ (n,m)$ of indices such that $ n \not\equal{}
m$ and \[ \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right).\]
[i]V. T. Sos[/i]
1959 Miklós Schweitzer, 10
[b]10.[/b] Prove that if a graph with $2n+1$ vertices has at least $3n+1$ edges, then the graph contains a circuit having an even number of edges. Prove further that this statemente does not hold for $3n$ edges. (By a circuit, we mean a closed line which does not intersect itself.) [b](C. 5)[/b]
2016 Korea USCM, 1
Find the following limit.
\[\lim_{n\to\infty} \frac{1}{n} \log \left(\sum_{k=2}^{2^n} k^{1/n^2} \right)\]
ICMC 2, 3
A ‘magic square’ of size \(n\) is an \(n\times n\) array of real numbers such that all the rows, all the columns and the two main diagonals have the same sum. Determine the dimension, over \(\mathbb{R}\), of the vector space of \(n\times n\) magic squares.\\
2015 IMC, 3
Let $F(0)=0$, $F(1)=\frac32$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$
for $n\ge2$.
Determine whether or not $\displaystyle{\sum_{n=0}^{\infty}\,
\frac{1}{F(2^n)}}$ is a rational number.
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
2020 IMC, 1
Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties.
1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$
2. $w$ contains an even number of $a$'s
3. $w$ contains an even number of $b$'s.
For example, for $n=2$ there are $6$ such words: $aa, bb, cc, dd, cd, dc.$
2012 IMC, 3
Is the set of positive integers $n$ such that $n!+1$ divides $(2012n)!$ finite or infinite?
[i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]
2008 Miklós Schweitzer, 6
Is it possible to draw circles on the plane so that every line intersects at least one of them but no more than $100$ of them?
ICMC 6, 1
The city of Atlantis is built on an island represented by $[ -1, 1]$, with skyline initially given by $f(x) = 1 - |x| $. The sea level is currently $y=0$, but due to global warming, it is rising at a rate of $0.01$ a year. For any position $-1 < x < 1$, while the building at $x$ is not completely submerged, then it is instantaneously being built upward at a rate of $r$ per year, where $r$ is the distance (along the $x$-axis) from this building to the nearest completely submerged building.
How long will it be until Atlantis becomes completely submerged?
[i]Proposed by Ethan Tan[/i]
ICMC 3, 4
Let n be a non-negative integer. Define the [i]decimal digit product[/i] \(D(n)\) inductively as follows:
- If \(n\) has a single decimal digit, then let \(D(n) = n\).
- Otherwise let \(D(n) = D(m)\), where \(m\) is the product of the decimal digits of \(n\).
Let \(P_k(1)\) be the probability that \(D(i) = 1\) where \(i\) is chosen uniformly randomly from the set of integers between 1 and \(k\) (inclusive) whose decimal digit products are not 0.
Compute \(\displaystyle\lim_{k\to\infty} P_k(1)\).
[i]proposed by the ICMC Problem Committee[/i]
2000 Miklós Schweitzer, 9
Let $M$ be a closed, orientable $3$-dimensional differentiable manifold, and let $G$ be a finite group of orientation preserving diffeomorphisms of $M$. Let $P$ and $Q$ denote the set of those points of $M$ whose stabilizer is nontrivial (that is, contains a nonidentity element of $G$) and noncyclic, respectively. Let $\chi (P)$ denote the Euler characteristic of $P$. Prove that the order of $G$ divides $\chi (P)$, and $Q$ is the union of $-2\frac{\chi(P)}{|G|}$ orbits of $G$.
2001 Putnam, 4
Let $S$ denote the set of rational numbers different from $ \{ -1, 0, 1 \} $. Define $f: S \rightarrow S $ by $f(x)=x-1/x$. Prove or disprove that \[ \cap_{n=1}^{\infty} f^{(n)} (S) = \emptyset \] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.
2007 Miklós Schweitzer, 8
For an $A=\{ a_i\}^{\infty}_{i=0}$ sequence let $SA=\{ a_0, a_0+a_1, a_0+a_1+a_2, \ldots\}$ be the sequence of partial sums of the $a_0+a_1+\ldots$ series. Does there exist a non-identically zero sequence $A$ such that all of the sequences $A, SA, SSA, SSSA, \ldots$ are convergent?
(translated by Miklós Maróti)