Found problems: 876
1989 Putnam, B2
Let S be a non-empty set with an associative operation that is left and right cancellative (xy=xz implies y=z, and yx = zx implies y = z). Assume that for every a in S the set {a^n : n = 0,1,2...} is finite. Must S be a group?
I haven't had much group theory at this point...
2008 Miklós Schweitzer, 9
For a given $\alpha >0$ let us consider the regular, non-vanishing $f(z)$ maps on the unit disc $\{ |z|<1 \}$ for which $f(0)=1$ and $\mathrm{Re}\, z\frac{f'(z)}{f(z)}>-\alpha$ ($|z|<1$). Show that the range of
$$g(z)=\frac{1}{(1-z)^{2\alpha}}$$
contains the range of all other such functions. Here we consider that regular branch of $g(z)$ for which $g(0)=1$.
(translated by Miklós Maróti)
MIPT student olimpiad autumn 2024, 2
$A,B \in M_{2\times 2}(C)$
Prove that:
$Tr(AAABBABAABBB)=tr(BBBAABABBAAA)$
2011 Putnam, B6
Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\},$
\[\sum_{k=0}^{p-1}k!n^k \quad \text{is not divisible by }p.\]
ICMC 2, 3
Show that if the faces of a tetrahedron have the same area, then they are congruent.
2001 Putnam, 5
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
2007 Putnam, 6
A [i]triangulation[/i] $ \mathcal{T}$ of a polygon $ P$ is a finite collection of triangles whose union is $ P,$ and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side of $ P$ is a side of exactly one triangle in $ \mathcal{T}.$ Say that $ \mathcal{T}$ is [i]admissible[/i] if every internal vertex is shared by $ 6$ or more triangles. For example
[asy]
size(100);
dot(dir(-100)^^dir(230)^^dir(160)^^dir(100)^^dir(50)^^dir(5)^^dir(-55));
draw(dir(-100)--dir(230)--dir(160)--dir(100)--dir(50)--dir(5)--dir(-55)--cycle);
pair A = (0,-0.25);
dot(A);
draw(A--dir(-100)^^A--dir(230)^^A--dir(160)^^A--dir(100)^^A--dir(5)^^A--dir(-55)^^dir(5)--dir(100));
[/asy]
Prove that there is an integer $ M_n,$ depending only on $ n,$ such that any admissible triangulation of a polygon $ P$ with $ n$ sides has at most $ M_n$ triangles.
2008 IMC, 6
For a permutation $ \sigma\in S_n$ with $ (1,2,\dots,n)\mapsto(i_1,i_2,\dots,i_n)$, define
\[ D(\sigma) \equal{} \sum_{k \equal{} 1}^n |i_k \minus{} k|
\]
Let
\[ Q(n,d) \equal{} \left|\left\{\sigma\in S_n : D(\sigma) \equal{} d\right\}\right|
\]
Show that when $ d \geq 2n$, $ Q(n,d)$ is an even number.
2012 IMC, 1
For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$ because
\[4=3+1=2+2=2+1+1=1+1+1.\]
Also define $p(0)=1$.
Prove that $p(n)-p(n-1)$ is the number of ways to express $n$ as a sum of integers each of which is strictly greater than 1.
[i]Proposed by Fedor Duzhin, Nanyang Technological University.[/i]
2019 Korea USCM, 6
A function $f:[0,\infty)\to[0,\infty)$ is integrable and
$$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$
Prove the following inequality.
$$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$
2021 Brazil Undergrad MO, Problem 6
We recursively define a set of [i]goody pairs[/i] of words on the alphabet $\{a,b\}$ as follows:
- $(a,b)$ is a goody pair;
- $(\alpha, \beta) \not= (a,b)$ is a goody pair if and only if there is a goody pair $(u,v)$ such that $(\alpha, \beta) = (uv,v)$ or $(\alpha, \beta) = (u,uv)$
Show that if $(\alpha, \beta)$ is a good pair then there exists a palindrome $\gamma$ (possibly empty) such that $\alpha\beta = a \gamma b$
2019 IMC, 5
Determine whether there exist an odd positive integer $n$ and $n\times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions:
[list=1]
[*]$\det (B)=1$;[/*]
[*]$AB=BA$;[/*]
[*]$A^4+4A^2B^2+16B^4=2019I$.[/*]
[/list]
(Here $I$ denotes the $n\times n$ identity matrix.)
[i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]
2022 District Olympiad, P1
Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions which satisfy \[\inf_{x>a}f(x)=g(a)\text{ and }\sup_{x<a}g(x)=f(a),\]for all $a\in\mathbb{R}.$ Given that $f$ has Darboux's Property (intermediate value property), show that functions $f$ and $g$ are continuous and equal to each other.
[i]Mathematical Gazette [/i]
2021 Alibaba Global Math Competition, 11
Let $M$ be a compact orientable $2n$-manifold with boundary, where $n \ge 2$. Suppose that $H_0(M;\mathbb{Q}) \cong \mathbb{Q}$ and $H_i(M;\mathbb{Q})=0$ for $i>0$. Prove that the order of $H_{n-1}(\partial M; \mathbb{Z})$ is a square number.
2019 Korea USCM, 4
For any $n\times n$ unitary matrices $A,B$, prove that $|\det (A+2B)|\leq 3^n$.
1979 Putnam, A1
Find positive integers $n$ and $a_1, a_2, \dots, a_n$ such that $$a_1+a_2+\dots a_n=1979$$ and the product $a_1a_2\dots a_n$ as large as possible.
1971 Putnam, A2
Determine all polynomials $P(x)$ such that $P(x^2+1)=(P(x))^2+1$ and $P(0)=0.$
1978 Miklós Schweitzer, 3
Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $ \sum_{i\equal{}1}^n 1/a_i \leq 1$. Let $ y$ denote the number of positive integers smaller that $ x$ not divisible by any of the $ a_i$. Prove that \[ y > \frac{cx}{\log x}\] with a suitable positive constant $ c$ (independent of $ x$ and the numbers $ a_i$).
[i]I. Z. Ruzsa[/i]
ICMC 5, 1
Let $T_n$ be the number of non-congruent triangles with positive area and integer side lengths summing to $n$. Prove that $T_{2022}=T_{2019}$.
[i]Proposed by Constantinos Papachristoforou[/i]
2003 Miklós Schweitzer, 5
Let $d>1$ be integer and $0<r<\frac12$. Show that there exist finitely many (depending only on $d,r$) nonzero vectors in $\mathbb{R}^d$ such that if the distance of a straight line in $\mathbb{R}^d$ from the integer lattice $\mathbb{Z}^d$ is at least $r$, then this line is orthogonal to one of these finitely many vectors.
(translated by L. Erdős)
2004 Putnam, A4
Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2\cdots x_n$ can be expressed identically in the form
\[x_1x_2\cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots +a_{in}x_n)^n\]
where the $c_i$ are rational numbers and each $a_{ij}$ is one of the numbers, $-1,0,1.$
ICMC 4, 2
Let \(A\) be a square matrix with entries in the field \(\mathbb Z / p \mathbb Z\) such that \(A^n - I\) is invertible for every positive integer \(n\). Prove that there exists a positive integer \(m\) such that \(A^m = 0\).
[i](A matrix having entries in the field \(\mathbb Z / p \mathbb Z\) means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of \(p\).)[/i]
[i]Proposed by Tony Wang[/i]
2014 Miklós Schweitzer, 9
Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula
\[\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.\]
Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.
ICMC 5, 5
A robot on the number line starts at $1$. During the first minute, the robot writes down the number $1$. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by $n/t$, where $n$ is the number it just moved to, and $t$ is the number of minutes elapsed. It then writes this number down. For example, if the robot moves right during the second minute, it would write down $2/2=1$.
Find the expected sum of all numbers it writes down, given that it is finite.
[i]Proposed by Ethan Tan[/i]
2021 Alibaba Global Math Competition, 20
Let $M=\bigoplus_{i \in \mathbb{Z}} \mathbb{C}e_i$ be an infinite dimensional $\mathbb{C}$-vector space, and let $\text{End}(M)$ denote the $\mathbb{C}$-algebra of $\mathbb{C}$-linear endomorphisms of $M$. Let $A$ and $B$ be two commuting elements in $\text{End}(M)$ satisfying the following condition: there exist integers $m \le n<0<p \le q$ satisfying $\text{gd}(-m,p)=\text{gcd}(-n,q)=1$, and such that for every $j \in \mathbb{Z}$, one has
\[Ae_j=\sum_{i=j+m}^{j+n} a_{i,j}e_i, \quad \text{with } a_{i,j} \in \mathbb{C}, a_{j+m,j}a_{j+n,j} \ne 0,\]
\[Be_j=\sum_{i=j+p}^{j+q} b_{i,j}e_i, \quad \text{with } b_{i,j} \in \mathbb{C}, b_{j+p,j}b_{j+q,j} \ne 0.\]
Let $R \subset \text{End}(M)$ be the $\mathbb{C}$-subalgebra generated by $A$ and $B$. Note that $R$ is commutative and $M$ can be regarded as an $R$-module.
(a) Show that $R$ is an integral domain and is isomorphic to $R \cong \mathbb{C}[x,y]/f(x,y)$, where $f(x,y)$ is a non-zero polynomial such that $f(A,B)=0$.
(b) Let $K$ be the fractional field of $R$. Show that $M \otimes_R K$ is a $1$-dimensional vector space over $K$.