Olympiad problems

ICMC 3, 4

Tags: college contests

Let $\mathcal{S}=\left\{S_1,S_2,\ldots,S_n\right\}$ be a set of $n\geq 2020$ distinct points on the Euclidean plane, no three of which are collinear. Andy the ant starts at some point $S_{i_1}$ in $\mathcal{S}$ and wishes to visit a series of 2020 points $\left\{S_{i_1},S_{i_2},\ldots,S_{i_{2020}}\right\}\subseteq\mathcal{S}$ in order, such that $i_j>i_k$ whenever $j>k$. It is known that ants can only travel between points in $\mathcal{S}$ in straight lines, and that an ant's path can never self-intersect. Find a positive integer $n$ such that Andy can always fulfill his wish. (Lower n will be awarded more marks. Bounds for this problem may be used as a tie-breaker, should the need to do so arise.) [i]Proposed by the ICMC Problem Committee[/i]

1959 Miklós Schweitzer, 7

Tags: complex numbers , college contests , real analysis

[b]7.[/b] Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers tending to zero. Prove that there exists a sequence $(\epsilon_n)_{n=1}^{\infty}$ (where $\epsilon_n = +1$ or $-1$) such that the series $\sum_{n=1}^{\infty} \epsilon_n z_n$ is convergente. [b](F. 9)[/b]

1961 Miklós Schweitzer, 10

Tags: college contests

[b]10.[/b] Given a straight line $g$ in the plane and a point $O$ on $g$. Construct, without making use of the Parallel Axiom, the half-line perpendicular to $g$ at the point $O$ and lying in one of the half-planes defined by $g$, under the following restrictions: The construction must be effected by use of a ruler and of a length standard (i.e. an etalon-segment) only; moreover, all lines and points of the construction must lie in the chosen half-plane. [b](G. 20)[/b]

1977 Putnam, B5

Tags: college contests

Suppose that $a_1,a_2,\dots a_n$ are real $(n>1)$ and $$A+ \sum_{i=1}^{n} a^2_i< \frac{1}{n-1} (\sum_{i=1}^{n} a_i)^2.$$ Prove that $A<2a_ia_j$ for $1\leq i<j\leq n.$

2005 Putnam, B4

Tags: Putnam , function , induction , calculus , integration , college contests

For positive integers $ m$ and $ n$, let $ f\left(m,n\right)$ denote the number of $ n$-tuples $ \left(x_1,x_2,\dots,x_n\right)$ of integers such that $ \left|x_1\right| \plus{} \left|x_2\right| \plus{} \cdots \plus{} \left|x_n\right|\le m$. Show that $ f\left(m,n\right) \equal{} f\left(n,m\right)$.

2003 Miklós Schweitzer, 1

Tags: college contests , Miklos Schweitzer , combinatorics , set theory

Let $(X, <)$ be an arbitrary ordered set. Show that the elements of $X$ can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour. (translated by L. Erdős)

2016 VJIMC, 3

Tags: linear algebra , matrix , college contests , eigenvalue

For $n \geq 3$ find the eigenvalues (with their multiplicities) of the $n \times n$ matrix $$\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\ 0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\ 1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\ 0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\ 0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1 \end{bmatrix}$$

1979 Putnam, B6

Tags: college contests

For $k=1,2 \dots, n$ let $z_k=x_k+iy_k,$ where the $x_k$ and $y_k$ are real and $i=\sqrt{-1}$. Let $r$ bet the absolute value of the real part of $$\pm \sqrt{z_1^2+z_2^2+\dots z_n^2}.$$ Prove that $r\leq |x_1|+|x_2|+ \dots +|x_n|.$

2004 Miklós Schweitzer, 5

Tags: college contests , Miklos Schweitzer , probability

Let $G$ be a non-solvable finite group and let $\varepsilon > 0$. Show that there exist a positive integer $k$ and a word $w\in F_k$ such that $w$ assumes the value $1$ with probability less than $\varepsilon$ when its $k$ arguments are considered to be independent and uniformly distributed random variables with values in $G$. (We write $F_k$ for the free group generated by $k$ elements.)

2000 Putnam, 6

Tags: Putnam , inequalities , college contests , combinatorics , combinatorial geometry , Probabilistic Method

Let $B$ be a set of more than $\tfrac{2^{n+1}}{n}$ distinct points with coordinates of the form $(\pm 1, \pm 1, \cdots, \pm 1)$ in $n$-dimensional space with $n \ge 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle.

2007 IMC, 1

Tags: geometry , geometric transformation , logarithms , function , rotation , IMC , college contests

Let $ f : \mathbb{R}\to \mathbb{R}$ be a continuous function. Suppose that for any $ c > 0$, the graph of $ f$ can be moved to the graph of $ cf$ using only a translation or a rotation. Does this imply that $ f(x) = ax+b$ for some real numbers $ a$ and $ b$?

2019 IMC, 6

Tags: IMC , differentiable function , function , real analysis , college contests

Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$. Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$. [i]Proposed by Fereshteh Malek, K. N. Toosi University of Technology[/i]

2010 Putnam, A2

Tags: Putnam , function , integration , college contests , real analysis , slope

Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$

2002 Putnam, 5

Tags: Putnam , college contests

A palindrome in base $b$ is a positive integer whose base-$b$ digits read the same backwards and forwards; for example, $2002$ is a $4$-digit palindrome in base $10$. Note that $200$ is not a palindrome in base $10$, but it is a $3$-digit palindrome: $242$ in base $9$, and $404$ in base $7$. Prove that there is an integer which is a $3$-digit palindrome in base $b$ for at least $2002$ different values of $b$.

1971 Putnam, B4

Tags: college contests

A "spherical ellipse" with foci $A,B$ on a given sphere is defined as the set of all points $P$ on the sphere such that $\overset{\Large\frown}{PA}+\overset{\Large\frown}{PB}=$ constant. Here $\overset{\Large\frown}{PA}$ denotes the shortest distance on the sphere between $P$ and $A$. Determine the entire class of real spherical ellipses which are circles.

1960 Miklós Schweitzer, 3

Tags: college contests

[b]3.[/b] Let $f(z)$ with $f(0)=1$ be regular in the unit disk and let $\left [\frac{\partial^2 \mid f(z)\mid}{\partial x\partial y} \right ] _{z=0} =1$. Show thatthe area of the image of the unit disk by $w= f(z)$ (taken with multiplicity) is not less than $\frac {1} {2}$ .[b](f. 6)[/b]

2005 IMC, 6

Tags: group theory , abstract algebra , IMC , college contests

6) $G$ group, $G_{m}$ and $G_{n}$ commutative subgroups being the $m$ and $n$ th powers of the elements in $G$. Prove $G_{gcd(m,n)}$ is commutative.

2012 IMC, 4

Tags: function , limit , integration , IMC , college contests

Let $f:\;\mathbb{R}\to\mathbb{R}$ be a continuously differentiable function that satisfies $f'(t)>f(f(t))$ for all $t\in\mathbb{R}$. Prove that $f(f(f(t)))\le0$ for all $t\ge0$. [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]

ICMC 6, 3

Tags: ICMC , college contests , probability , combinatorics , random process

The numbers $1, 2, \dots , n$ are written on a blackboard and then erased via the following process:[list] [*] Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. [*] Each minute for the next $n -1$ minutes, a pair of numbers still on the blackboard is chosen uniformly at random and the smaller one is erased. [*] In minute $n$, the last number is erased. [/list] What is the probability that the smaller circled number is erased before the larger? [i]Proposed by Ethan Tan[/i]

2010 Miklós Schweitzer, 7

Tags: college contests , Miklos Schweitzer , real analysis

Is there any sequence $(a_n)_{n=1}^{\infty}$ of non-negative numbers, for which $\sum_{n=1}^{\infty} a_n^2<\infty$ , but $\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}\frac{a_{kn}}{k} \right)^2=\infty$ ? [hide=Remark]That contest - Miklos Schweitzer 2010- is missing on the contest page here for now being. The statements of all problems that year can be found [url=http://www.math.u-szeged.hu/~mmaroti/schweitzer/]here[/url], but unfortunately only in Hungarian. I tried google translate but it was a mess. So, it would be wonderful if someone knows Hungarian and wish to translate it. [/hide]

2015 Kyoto University Entry Examination, 2

Tags: geometry , college contests

2. Find the minimum area of quadrilateral satisfy two condition as follows, (a) At least two interior angles are right angles. (b) A circle radius of $1$ inscribed.

1977 Putnam, B2

Tags: college contests

Given a convex quadrilateral $ABCD$ and a point $O$ not in the plane $ABCD$, locate point $A'$ on line $OA,$ point $B'$ on the line $OB$, point $C'$ on line $OC,$ and point $D'$ on line $OD$ so that $A'B'C'D'$ is a parallelogram.

1996 Putnam, 5

Tags: Putnam , floor function , college contests

Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ $\emph{balanced}$ if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.

2019 IMC, 4

Tags: IMC , recurrence relation , Sequences , Sequence , college contests , generating functions

Let $(n+3)a_{n+2}=(6n+9)a_{n+1}-na_n$ and $a_0=1$ and $a_1=2$ prove that all the terms of the sequence are integers

1957 Miklós Schweitzer, 10

Tags: abstract algebra , group theory , college contests

[b]10.[/b] An Abelian group $G$ is said to have the property $(A)$ if torsion subgroup of $G$ is a direct summand of $G$. Show that if $G$ is an Abelian group such that $nG$ has the property $(A)$ for some positive integer $n$, then $G$ itself has the property $(A)$. [b](A. 13)[/b]