Olympiad problems

1968 Putnam, A1

Tags: integration , algebra , polynomial , Putnam , calculus , college contests

Prove $ \ \ \ \frac{22}{7}\minus{}\pi \equal{}\int_0^1 \frac{x^4(1\minus{}x)^4}{1\plus{}x^2}\ dx$.

2014 IMC, 3

Tags: IMC , college contests , calculus , derivative , function

Let $f(x)=\frac{\sin x}{x}$, for $x>0$, and let $n$ be a positive integer. Prove that $|f^{(n)}(x)|<\frac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\mathrm{th}}$ derivative of $f$. (Proposed by Alexander Bolbot, State University, Novosibirsk)

2008 IMC, 5

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

Does there exist a finite group $ G$ with a normal subgroup $ H$ such that $ |\text{Aut } H| > |\text{Aut } G|$? Disprove or provide an example. Here the notation $ |\text{Aut } X|$ for some group $ X$ denotes the number of isomorphisms from $ X$ to itself.

1985 Miklós Schweitzer, 6

Tags: Miklos Schweitzer , college contests , group theory , abstract algebra , superior algebra

Determine all finite groups $G$ that have an automorphism $f$ such that $H\not\subseteq f(H)$ for all proper subgroups $H$ of $G$. [B. Kovacs]

2008 IMC, 5

Tags: linear algebra , matrix , IMC , college contests

Let $ n$ be a positive integer, and consider the matrix $ A \equal{} (a_{ij})_{1\leq i,j\leq n}$ where $ a_{ij} \equal{} 1$ if $ i\plus{}j$ is prime and $ a_{ij} \equal{} 0$ otherwise. Prove that $ |\det A| \equal{} k^2$ for some integer $ k$.

2015 Kyoto University Entry Examination, 4

Tags: college contests , 3D geometry

4. Consider spherical surface $S$ which radius is $1$, central point $(0,0,1)$ in $xyz$ space. If point $Q$ move to points on S expect $(0,0,2)$. Let $R$ be an intersection of plane $z=0$ and line $l$ pass point $Q$ and point $P (1,0,2)$. Find the range of moving $R$, then illustrate it.

2006 Putnam, B3

Tags: Putnam , rotation , induction , floor function , ceiling function , college contests

Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A\cup B=S,\ A\cap B=\emptyset,$ and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_{S}$ be the number of linear partitions of $S.$ For each positive integer $n,$ find the maximum of $L_{S}$ over all sets $S$ of $n$ points.

2021 Alibaba Global Math Competition, 16

Tags: group theory , abstract algebra , invariant , representation theory , Center , algebra , college contests

Let $G$ be a finite group, and let $H_1, H_2 \subset G$ be two subgroups. Suppose that for any representation of $G$ on a finite-dimensional complex vector space $V$, one has that \[\text{dim} V^{H_1}=\text{dim} V^{H_2},\] where $V^{H_i}$ is the subspace of $H_i$-invariant vectors in $V$ ($i=1,2$). Prove that \[Z(G) \cap H_1=Z(G) \cap H_2.\] Here $Z(G)$ denotes the center of $G$.

2003 Miklós Schweitzer, 9

Tags: college contests , Miklos Schweitzer , algebra , function , domain

Given finitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial $C(q)$ of degree two so that the following holds: for any $q\ge 1$ integer, if the half planes cover each point of the plane at least $q$ times, then the set of points covered exactly $q$ times is the union of at most $C(q)$ domains. (translated by L. Erdős)

2005 IMC, 1

Tags: linear algebra , matrix , IMC , college contests

Let $A$ be a $n\times n$ matrix such that $A_{ij} = i+j$. Find the rank of $A$. [hide="Remark"]Not asked in the contest: $A$ is diagonalisable since real symetric matrix it is not difficult to find its eigenvalues.[/hide]

2012 Miklós Schweitzer, 6

Tags: college contests

Let $A,B,C$ be matrices with complex elements such that $[A,B]=C, [B,C]=A$ and $[C,A]=B$, where $[X,Y]$ denotes the $XY-YX$ commutator of the matrices. Prove that $e^{4 \pi A}$ is the identity matrix.

2002 Putnam, 1

Tags: Putnam , probability , induction , conditional probability , college contests

Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly $50$ of her first $100$ shots?

2018 Miklós Schweitzer, 3

Tags: linear algebra , matrix , combinatorics , college contests , Miklos Schweitzer

We call an $n\times n$ matrix [i]well groomed[/i] if it only contains elements $0$ and $1$, and it does not contain the submatrix $\begin{pmatrix} 1& 0\\ 0 & 1 \end{pmatrix}.$ Show that there exists a constant $c>0$ such that every well groomed, $n\times n$ matrix contains a submatrix of size at least $cn\times cn$ such that all of the elements of the submatrix are equal. (A well groomed matrix may contain the submatrix $\begin{pmatrix} 0& 1\\ 1 & 0 \end{pmatrix}.$ )

1990 Putnam, B3

Tags: linear algebra , matrix , pigeonhole principle , Putnam , college contests

Let $S$ be a set of $ 2 \times 2 $ integer matrices whose entries $a_{ij}(1)$ are all squares of integers and, $(2)$ satisfy $a_{ij} \le 200$. Show that $S$ has more than $ 50387 (=15^4-15^2-15+2) $ elements, then it has two elements that commute.

1995 Putnam, 4

Tags: Putnam , induction , continued fraction , college contests

Evaluate : \[ \sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-\cdots}}} \] Express your expression in the form $\frac{a+b\sqrt{c}}{d}$ where $a,b,c,d\in \Bbb{Z}$.

2010 Putnam, B3

Tags: Putnam , number theory , greatest common divisor , pigeonhole principle , relatively prime , college contests

There are 2010 boxes labeled $B_1,B_2,\dots,B_{2010},$ and $2010n$ balls have been distributed among them, for some positive integer $n.$ You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving [i]exactly[/i] $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?

1994 Putnam, 4

Tags: Putnam , linear algebra , matrix , limit , induction , college contests

For $n\ge 1$ let $d_n$ be the $\gcd$ of the entries of $A^n-\mathcal{I}_2$ where \[ A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}\quad \text{ and }\quad \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix}\] Show that $\lim_{n\to \infty}d_n=\infty$.

ICMC 4, 1

Tags: ICMC , college contests , Sets , combinatorics , counting

Let $S$ be a set with 10 distinct elements. A set $T$ of subsets of $S$ (possibly containing the empty set) is called [i]union-closed[/i] if, for all $A, B \in T$, it is true that $A \cup B \in T$. Show that the number of union-closed sets $T$ is less than $2^{1023}$. [i]Proposed by Tony Wang[/i]

ICMC 3, 3

Tags: college contests

Let $\mathbb{R}$ denote the set of real numbers. A subset $S\subseteq\mathbb{R}$ is called [i]dense[/i] if any non-empty open interval of $\mathbb{R}$ contains at least one element in $S$. For a function $f:\mathbb{R}\to\mathbb{R}$, let $\mathcal{O}_f(x)$ denote the set $\left\{x,f(x),f(f(x)),\ldots\right\}$. (a) Is there a function $g:\mathbb{R}\to\mathbb{R}$, continuous everywhere in $\mathbb{R}$ such that $\mathcal{O}_g(x)$ is dense for all $x\in\mathbb{R}$ for all $x\in\mathbb{R}$? (b) Is there a function $h:\mathbb{R}\to\mathbb{R}$, continuous at all but a single $x_0\in\mathbb{R}$, such that $\mathcal{O}_h(x)$ is dense for all $x\in\mathbb{R}$? [i]Proposed by the ICMC Problem Committee[/i]

1966 Putnam, A2

Tags: college contests

Let $a,b,c$ be the lengths of the sides of a triangle, let $p=(a+b+c)/2$, and $r$ be the radius of the inscribed circle. Show that $$\frac{1}{(p-a)^2}+ \frac{1}{(p-b)^2}+\frac{1}{(p-c)^2} \geq \frac{1}{r^2}.$$

2003 Putnam, 3

Tags: Putnam , number theory , least common multiple , floor function , prime factorization , college contests

Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\] (Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)

2004 IMC, 3

Tags: inequalities , triangle inequality , IMC , college contests

Let $D$ be the closed unit disk in the plane, and let $z_1,z_2,\ldots,z_n$ be fixed points in $D$. Prove that there exists a point $z$ in $D$ such that the sum of the distances from $z$ to each of the $n$ points is greater or equal than $n$.

2003 Miklós Schweitzer, 8

Tags: college contests , Miklos Schweitzer , function , series

Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)? (translated by L. Erdős)

2002 Putnam, 1

Tags: Putnam , calculus , derivative , function , limit , college contests

Let $k$ be a fixed positive integer. The $n$th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$, where $P_n(x)$ is a polynomial. Find $P_n(1)$.

2009 IMC, 2

Tags: function , inequalities , IMC , college contests , Integral inequality

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a two times differentiable function satisfying $f(0)=1,f^{\prime}(0)=0$ and for all $x\in [0,\infty)$, it satisfies \[ f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0 \] Prove that, for all $x\in [0,\infty)$, \[ f(x)\ge 3e^{2x}-2e^{3x} \]