Found problems: 876
1976 Putnam, 4
For a point $P$ on an ellipse, let $d$ be the distance from the center of the ellipse to the line tangent to the ellipse at $P.$ Prove that $(PF_1)(PF_2)d^2$ is constant as $P$ varies on the ellipse, where $PF_1$ and $PF_2$ are distances from $P$ to the foci $F_1$ and $F_2$ of the ellipse.
1986 Miklós Schweitzer, 3
(a) Prove that for every natural number $k$, there are positive integers $a_1<a_2<\ldots <a_k$ such that $a_i-a_j$ divides $a_i$ for all $1\leq i, j\leq k, i\neq j$.
(b) Show that there is an absolute constant $C>0$ such that $a_1>k^{Ck}$ for every sequence $a_1,\ldots, a_k$ of numbers that satisfy the above divisibility condition.
[A. Balogh, I. Z. Ruzsa]
2008 Miklós Schweitzer, 11
Let $\zeta_1, \ldots, \zeta_n$ be (not necessarily independent) random variables with normal distribution for which $E\zeta_j=0$ and $E\zeta_j^2\le 1$ for all $1\le j\le n$. Prove that
$$E\left( \max_{1\le j\le n} \zeta_j \right)\le\sqrt{2\log n}$$
(translated by Miklós Maróti)
2004 Miklós Schweitzer, 4
Determine all totally multiplicative and non-negative functions $f\colon\mathbb{Z}\rightarrow \mathbb{Z}$ with the property that if $a, b\in \mathbb{Z}$ and $b\neq 0$, then there exist integers $q$ and $r$ such that $a-qb+r$ and $f(r)<f(b)$.
2009 IMC, 5
Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S\subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$.
Say that a vector subspace $T\subseteq \mathbb{M}$ is a $\emph{covering matrix space}$ if
\[ \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p \]
Such a $T$ is minimal if it doesn't contain a proper vector subspace $S\subset T$ such that $S$ is also a covering matrix space.
[list]
(a) (8 points) Let $T$ be a minimal covering matrix space and let $n=\dim (T)$
Prove that
\[ \delta(T)\le \dbinom{n}{2} \]
(b) (2 points) Prove that for every integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T=n$ and $\delta(T)=\dbinom{n}{2}$[/list]
2019 Korea USCM, 7
For a real number $a$ and an integer $n(\geq 2)$, define
$$S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}$$
Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real.
1985 Miklós Schweitzer, 1
[b]1.[/b] Some proper partitions $P_1, \dots , P_n$ of a finite set $S$ (that is, partitions containing at least two parts) are called [i]independent[/i] if no matter how we choose one class from each partition, the intersection of the chosen classes is nonempty. Show that if the inequality
$\frac{\left | S \right | }{2} < \left |P_1 \right | \dots \left |P_n \right |\qquad \quad (*)$
holds for some independent partitions, then $P_1, \dots , P_n$ is maximal in the sense that there is no partition $P$ such that $P,P_1, \dots , P_n$ are independent. On the other hand, show that inequality $(*)$ is not necessary for this maximality. ([b]C.20[/b])
[E. Gesztelyi]
2004 Putnam, B5
Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
1995 Putnam, 1
For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi^{\prime}$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi^{\prime}(x) = \pi^{\prime}(y)$.
2016 VJIMC, 1
Let $a,b,c$ be positive real numbers such that $a + b + c = 1$. Show that
$$\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728$$
ICMC 2, 6
A country has four political parties - the Blue Party, the Red Party, the Yellow Party and the Orange Party - and a parliament of 650 seats.
(a) How many ways are there to divide the seats among the four parties so that none of the parties have a majority? (To have a majority that party must hold more than half of the seats.)
The parliament is particularly worried about cyber security. They have decided that all login passwords must be of length exactly 6 and be a combination of a legal set of elements made up of the digits 0-9, the 52 upper and lower case letters (a-z and A-Z), and five special characters: \ICMC 2018/19 Round 1, Problem 1 $, £, *, &, %. For the password to be allowed, it must contain at least one letter or special character and any letter or special character in the password must be followed by a digit (so it must end in a digit).
(b) The Blue members of parliament have decided to choose their password by selecting 6 elements from the legal set without replacement. What is the probability it is allowed?
Note: you may leave your answers as combinatorial or factorial terms.
ICMC 3, 3
Consider a grid of points where each point is coloured either white or black, such that no two rows have the same sequence of colours and no two columns have the same sequence of colours. Let a [i]table[/i] denote four points on the grid that form the vertices of a rectangle with sides parallel to those of the grid. A table is called [i]balanced[/i] if one diagonal pair of points are coloured white and the other diagonal pair black.
Determine all possible values of \(k \geq 2\) for which there exists a colouring of a \(k\times 2019\) grid with no balanced tables.
[i]proposed by the ICMC Problem Committee[/i]
2017 IMC, 10
Let $K$ be an equilateral triangle in the plane. Prove that for every $p>0$ there exists an $\varepsilon>0$ with the following property: If $n$ is a positive integer, and $T_1,\ldots,T_n$ are non-overlapping triangles inside $K$ such that each of them is homothetic to $K$ with a negative ratio, and $$ \sum_{\ell=1}^n \textrm{area}(T_\ell) > \textrm{area}(K)-\varepsilon, $$ then $$ \sum_{\ell=1}^n \textrm{perimeter}(T_\ell) > p. $$
2019 Miklós Schweitzer, 7
Given a polynomial $P$, assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$.
1961 Miklós Schweitzer, 9
[b]9.[/b] Spin a regular coin repeatedly until the heads and tails turned up both reach the number $k$ ($k= 1, 2, \dots $); denote by $v_k$ the number of the necessary throws. Find the distribution of the random variable $v_k$ and the limit-distribution of the random variable $\frac {v_k -2k}{\sqrt {2k}}$ as $k \to \infty$. [b](P. 10)[/b]
2010 Contests, A1
Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same?
[When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]
2018 VTRMC, 5
For $n \in \mathbb{N}$, let $a_n = \int _0 ^{1/\sqrt{n}} | 1 + e^{it} + e^{2it} + \dots + e^{nit} | \ dt$. Determine whether the sequence $(a_n) = a_1, a_2, \dots$ is bounded.
2019 Brazil Undergrad MO, 1
Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$
with rational entries and size $ 2019 $ such that:
a) $ A ^ 3 + 6A ^ 2-2I = 0 $?
b) $ A ^ 4 + 6A ^ 3-2I = 0 $?
1976 Putnam, 5
Evaluate $$\sum_{k=0}^n (-1)^k \binom{n}{k} (x-k)^n.$$
1955 Miklós Schweitzer, 7
[b]7.[/b] Prove that for any odd prime number $p$, the polynomial
$2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$
is congruent mod $p$ to the square of a polynomial with integer coefficients. [b](N. 21)[/b]
*This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594
2023 SG Originals, Q4
Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square?
[i]Proposed by Dylan Toh[/i]
2006 Putnam, B4
Let $Z$ denote the set of points in $\mathbb{R}^{n}$ whose coordinates are $0$ or $1.$ (Thus $Z$ has $2^{n}$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^{n}$.) Given a vector subspace $V$ of $\mathbb{R}^{n},$ let $Z(V)$ denote the number of members of $Z$ that lie in $V.$ Let $k$ be given, $0\le k\le n.$ Find the maximum, over all vector subspaces $V\subseteq\mathbb{R}^{n}$ of dimension $k,$ of the number of points in $V\cap Z.$
2009 IMC, 3
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that
\[ A^2B+BA^2=2ABA \]
Prove there exists $k\in \mathbb{N}$ such that
\[ (AB-BA)^k=\mathbf{0}_n\]
Here $\mathbf{0}_n$ is the null matrix of order $n$.
ICMC 2, 2
In the symmetric group \(S_n\ (n \geq 3)\), let \(G_{a,b}\) be the subgroup generated by the 2-cycle \((a\ b)\) and the n-cycle \((1\ 2\ \cdots\ n)\). Find the index \(\left|S_n : G_{a,b}\right|\).
2008 Miklós Schweitzer, 10
Let $V$ be the set of non-collinear pairs of vectors in $\mathbb{R}^3$, and $H$ be the set of lines passing through the origin. Is is true that for every continuous map $f\colon V\rightarrow H$ there exists a continuous map $g\colon V\rightarrow \mathbb{R}^3\,\backslash\,\{ 0\}$ such that $g(v)\in f(v)$ for all $v\in V$?
(translated by Miklós Maróti)