Olympiad problems

2016 IMC, 5

Let $A$ be a $n\times n$ complex matrix whose eigenvalues have absolute value at most $1$. Prove that $$ \|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}. $$ (Here $\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|$ for every $n\times n$ matrix $B$ and $\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}$ for every complex vector $x\in\mathbb{C}^n$.) (Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University)

2018 Brazil Undergrad MO, 12

Tags: probability , geometry , Brazilian Math Olympiad , college contests

Let $ABC$ be an equilateral triangle. $A $ point $P$ is chosen at random within this triangle. What is the probability that the sum of the distances from point $P$ to the sides of triangle $ABC$ are measures of the sides of a triangle?

1960 Miklós Schweitzer, 7

Tags: college contests , real analysis , Functional Analysis

[b]7.[/b] Define the generalized derivative at $x_0$ of the function $f(x)$ by $\lim_{h \to 0} 2 \frac{ \frac{1}{h} \int_{x_0}^{x_0+h} f(t) dt - f(x_0)}{h}$ Show that there exists a function, continuous everywhere, which is nowhere differentiable in this general sense [b]( R. 8)[/b]

1954 Miklós Schweitzer, 7

Tags: group theory , abstract algebra , college contests

[b]7.[/b] Find the finite groups having only one proper maximal subgroup. [b](A.12)[/b]

2021 Alibaba Global Math Competition, 17

Tags: algebra , polynomial , number theory , college contests , finite field

Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by \[\tau(f)(x)=f(x+1).\] Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$.

1990 Putnam, B2

Tags: induction , Putnam , college contests

Prove that for $ |x| < 1 $, $ |z| > 1 $, \[ 1 + \displaystyle\sum_{j=1}^{\infty} \left( 1 + x^j \right) P_j = 0, \]where $P_j$ is \[ \dfrac {(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3)\cdots(z-x^j)}. \]

2016 IMC, 5

Tags: IMC , IMC 2016 , prime numbers , permutations , college contests

Let $S_n$ denote the set of permutations of the sequence $(1,2,\dots, n)$. For every permutation $\pi=(\pi_1, \dots, \pi_n)\in S_n$, let $\mathrm{inv}(\pi)$ be the number of pairs $1\le i < j \le n$ with $\pi_i>\pi_j$; i. e. the number of inversions in $\pi$. Denote by $f(n)$ the number of permutations $\pi\in S_n$ for which $\mathrm{inv}(\pi)$ is divisible by $n+1$. Prove that there exist infinitely many primes $p$ such that $f(p-1)>\frac{(p-1)!}{p}$, and infinitely many primes $p$ such that $f(p-1)<\frac{(p-1)!}{p}$. (Proposed by Fedor Petrov, St. Petersburg State University)

2004 IMC, 4

Tags: linear algebra , matrix , algebra , polynomial , vector , IMC , college contests

For $n\geq 1$ let $M$ be an $n\times n$ complex array with distinct eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_k$, with multiplicities $m_1,m_2,\ldots,m_k$ respectively. Consider the linear operator $L_M$ defined by $L_MX=MX+XM^T$, for any complex $n\times n$ array $X$. Find its eigenvalues and their multiplicities. ($M^T$ denotes the transpose matrix of $M$).

2019 Miklós Schweitzer, 8

Tags: real analysis , college contests , Miklos Schweitzer

Let $f: \mathbb{R} \to \mathbb{R}$ be a measurable function such that $f(x+t) - f(x)$ is locally integrable for every $t$ as a function of $x$. Prove that $f$ is locally integrable.

2002 Putnam, 4

Tags: Putnam , modular arithmetic , college contests

An integer $n$, unknown to you, has been randomly chosen in the interval $[1,2002]$ with uniform probability. Your objective is to select $n$ in an ODD number of guess. After each incorrect guess, you are informed whether $n$ is higher or lower, and you $\textbf{must}$ guess an integer on your next turn among the numbers that are still feasibly correct. Show that you have a strategy so that the chance of winning is greater than $\tfrac{2}{3}$.

1977 Putnam, A4

Tags: college contests

For $0<x<1,$ express $$\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$$ as a rational function of $x.$

1999 Putnam, 5

Tags: Putnam , polynomial , integration , calculus , inequalities , college contests , real analysis

Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]

2012 Miklós Schweitzer, 8

Tags: college contests , Miklos Schweitzer

For any function $f: \mathbb{R}^2\to \mathbb{R}$ consider the function $\Phi_f:\mathbb{R}^2\to [-\infty,\infty]$ for which $\Phi_f(x,y)=\limsup_{ z \to y} f(x,z)$ for any $(x,y) \in \mathbb{R}^2$. [list=a] [*]Is it true that if $f$ is Lebesgue measurable then $\Phi_f$ is also Lebesgue measurable?[/*] [*]Is it true that if $f$ is Borel measurable then $\Phi_f$ is also Borel measurable?[/*] [/list]

2013 Putnam, 3

Tags: Putnam , function , induction , linear algebra , matrix , college contests , Putnam 2013

Let $P$ be a nonempty collection of subsets of $\{1,\dots,n\}$ such that: (i) if $S,S'\in P,$ then $S\cup S'\in P$ and $S\cap S'\in P,$ and (ii) if $S\in P$ and $S\ne\emptyset,$ then there is a subset $T\subset S$ such that $T\in P$ and $T$ contains exactly one fewer element than $S.$ Suppose that $f:P\to\mathbb{R}$ is a function such that $f(\emptyset)=0$ and \[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\] Must there exist real numbers $f_1,\dots,f_n$ such that \[f(S)=\sum_{i\in S}f_i\] for every $S\in P?$

2024 CIIM, 5

Tags: combinatorics , Coloring , limit , college contests

A board of size $3 \times N$ initially has all of its cells painted white. Let $a(N)$ be the maximum number of cells that can be painted black in such a way that no three consecutive cells (either horizontally, vertically, or diagonally) are painted black. Prove that \[ \lim_{N \to \infty} \frac{a(N)}{N} \] exists and determine its value.

1977 Putnam, B6

Tags: college contests

Let $H$ be a subgroup with $h$ elements in a group $G.$ Suppose that $G$ has an element $a$ such that for all $x$ in $H,$ $(xa)^3=1,$ the identity. In $G$, let $P$ be the subset of all products $x_1ax_2a\dots x_na,$ with $n$ a positive integer and the $x_i$ in $H.$ (a) Show that $P$ is a finite set. (b) Show that, in fact, $P$ has no more that $3h^2$ elements.

ICMC 6, 3

Tags: ICMC , combinatorics , Fibonacci , college contests

Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started. [i]Proposed by Dylan Toh[/i]

2002 Miklós Schweitzer, 1

Tags: college contests , Miklos Schweitzer , function

For an arbitrary ordinal number $\alpha$ let $H(\alpha)$ denote the set of functions $f\colon \alpha \rightarrow \{ -1,0,1\}$ that map all but finitely many elements of $\alpha$ to $0$. Order $H(\alpha)$ according to the last difference, that is, for $f, g\in H(\alpha)$ let $f\prec g$ if $f(\beta) < g(\beta)$ holds for the maximum ordinal number $\beta < \alpha$ with $f(\beta) \neq g(\beta)$. Prove that the ordered set $(H(\alpha), \prec)$ is scattered (i.e. it doesn't contain a subset isomorphic to the set of rational numbers with the usual order), and that any scattered order type can be embedded into some $(H(\alpha), \prec)$.

2016 IMC, 3

Tags: IMC , IMC 2016 , college contests , abstract algebra , function

Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ satisfying the following three properties: (i) $f(x)\neq x$, (ii) $f(f(x))=x$, (iii) $f(f(f(x+1)+1)+1)=x$ for all $x\in\mathbb{Z}_n$. Prove that $n\equiv 2 \pmod4$. (Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)

ICMC 4, 6

Tags: combinatorics , ICMC , college contests

There are $n+1$ squares in a row, labelled from 0 to $n$. Tony starts with $k$ stones on square 0. On each move, he may choose a stone and advance the stone up to $m$ squares where $m$ is the number of stones on the same square (including itself) or behind it. Tony's goal is to get all stones to square $n$. Show that Tony cannot achieve his goal in fewer than $\frac{n}{1} + \frac{n}{2} + \cdots + \frac{n}{k}$ moves. [i]Proposed by Tony Wang[/i]

2014 Paenza, 3

Tags: algebra , polynomial , Vieta , college contests

Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.

2007 Putnam, 4

Tags: Putnam , algebra , polynomial , college contests

Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that \[ (P(X))^2\plus{}(Q(X))^2\equal{}X^{2n}\plus{}1\] and $ \text{deg}P<\text{deg}{Q}.$

2021 Alibaba Global Math Competition, 7

Tags: Convergence , sobolev space , continuity , college contests

A subset $Q \subset H^s(\mathbb{R})$ is said to be equicontinuous if for any $\varepsilon>0$, $\exists \delta>0$ such that \[\|f(x+h)-f(x)\|_{H^s}<\varepsilon, \quad \forall \vert h\vert<\delta, \quad f \in Q.\] Fix $r<s$, given a bounded sequence of functions $f_n \in H^s(\mathbb{R}$. If $f_n$ converges in $H^r(\mathbb{R})$ and equicontinuous in $H^s(\mathbb{R})$, show that it also converges in $H^s(\mathbb{R})$.

1966 Putnam, A5

Tags: college contests

Let $C$ denote the family of continuous functions on the real axis. Let $T$ be a mapping of $C$ into $C$ which has the following properties: 1. $T$ is linear, i.e. $T(c_1\psi _1+c_2\psi _2)= c_1T\psi _1+c_2T\psi _2$ for $c_1$ and $c_2$ real and $\psi_1$ and $\psi_2$ in $C$. 2. $T$ is local, i.e. if $\psi_1 \equiv \psi_2$ in some interval $I$ then also $T\psi_1 \equiv T\psi_2$ holds in $I$. Show that $T$ must necessarily be of the form $T\psi(x)=f(x)\psi(x)$ where $f(x)$ is a suitable continuous function.

ICMC 5, 3

Tags: linear algebra , college contests , Determinants , ICMC

Let $\mathcal M$ be the set of $n\times n$ matrices with integer entries. Find all $A\in\mathcal M$ such that $\det(A+B)+\det(B)$ is even for all $B\in\mathcal M$. [i]Proposed by Ethan Tan[/i]