Found problems: 876
2014 Paenza, 1
Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation:
\[a_{n+1}=10^n a_n^2\]
(a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$.
(b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.
1957 Miklós Schweitzer, 3
[b]3.[/b] Let $A$ be a subset of n-dimensional space containing at least one inner point and suppose that, for every point pair $x, y \in A$, the subset $A$ contains the mid point of the line segment beteween $x$ and $y$. Show that $A$ consists of a convex open set and of some of its boundary points. [b](St. 1)[/b]
2020 Putnam, A2
Let $k$ be a nonnegative integer. Evaluate
\[ \sum_{j=0}^k 2^{k-j} \binom{k+j}{j}. \]
2018 Brazil Undergrad MO, 21
Consider $ p (x) = x ^ n + a_ {n-1} x ^ {n-1} + ... + a_ {1} x + 1 $ a polynomial of positive real coefficients, degree $ n \geq 2 $ e with $ n $ real roots. Which of the following statements is always true?
a) $ p (2) <2 (2 ^ {n-1} +1) $ (b) $ p (1) <3 $ c) $ p (1)> 2 ^ n $ d) $ p (3 ) <3 (2 ^ {n-1} -2) $
1985 Miklós Schweitzer, 11
Let $\xi (E, \pi, B)\, (\pi\colon E\rightarrow B)$ be a real vector bundle of finite rank, and let
$$\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)$$
be the tangent bundle of $E$, where $V\xi=\mathrm{Ker}\, d\pi$ is the vertical subbundle of $\tau_E$. Let us denote the projection operators corresponding to the splitting $(*)$ by $v$ and $h$. Construct a linear connection $\nabla$ on $V\xi$ such that
$$\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]$$
($X$ and $Y$ are vector fields on $E$, $[.,\, .]$ is the Lie bracket, and all data are of class $\mathcal C^\infty$. [J. Szilasi]
1997 Putnam, 4
Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^n$. Prove the inequality for all integers $k\ge 0$ :
\[ 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1 \]
1999 Putnam, 2
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.
2018 CIIM, Problem 6
Let $\{x_n\}$ be a sequence of real numbers in the interval $[0,1)$. Prove that there exists a sequence $1 < n_1 < n_2 < n_3 < \cdots$ of positive integers such that the following limit exists $$\lim_{i,j \to \infty} x_{n_i+n_j}. $$
That is, there exists a real number $L$ such that for every $\epsilon > 0,$ there exists a positive integer $N$ such that if $i,j > N$, then $|x_{n_i+n_j}-L| < \epsilon.$
2007 IMC, 2
Let $ n\ge 2$ be an integer. What is the minimal and maximal possible rank of an $ n\times n$ matrix whose $ n^{2}$ entries are precisely the numbers $ 1, 2, \ldots, n^{2}$?
1957 Miklós Schweitzer, 9
[b]9.[/b] Find all pairs of linear polynomials $f(x)$, $g(x)$ with integer coefficients for which there exist two polynomials $u(x)$, $v(x)$ with integer coefficients such that $f(x)u(x)+g(x)v(x)=1$. [b](A. 8)[/b]
1996 Putnam, 1
Find the least number $A$ such that for any two squares of combined area $1$, a rectangle of area $A$ exists such that the two squares can be packed in the rectangle (without the interiors of the squares overlapping) . You may assume the sides of the squares will be parallel to the sides of the rectangle.
2005 Putnam, A2
Let $S=\{(a,b)|a=1,2,\dots,n,b=1,2,3\}$. A [i]rook tour[/i] of $S$ is a polygonal path made up of line segments connecting points $p_1,p_2,\dots,p_{3n}$ is sequence such that
(i) $p_i\in S,$
(ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1\le i<3n,$
(iii) for each $p\in S$ there is a unique $i$ such that $p_i=p.$
How many rook tours are there that begin at $(1,1)$ and end at $(n,1)?$
(The official statement includes a picture depicting an example of a rook tour for $n=5.$ This example consists of line segments with vertices at which there is a change of direction at the following points, in order: $(1,1),(2,1),(2,2),(1,2), (1,3),(3,3),(3,1),(4,1), (4,3),(5,3),(5,1).$)
1958 Miklós Schweitzer, 2
[b]2.[/b] Let $A(x)$ denote the number of positive integers $n$ not greater than $x$ and having at least one prime divisor greater than $\sqrt[3]{n}$. Prove that $\lim_{x\to \infty} \frac {A(x)}{x}$ exists. [b](N. 15)[/b]
1983 Putnam, B6
Let $ k$ be a positive integer, let $ m\equal{}2^k\plus{}1$, and let $ r\neq 1$ be a complex root of $ z^m\minus{}1\equal{}0$. Prove that there exist polynomials $ P(z)$ and $ Q(z)$ with integer coefficients such that $ (P(r))^2\plus{}(Q(r))^2\equal{}\minus{}1$.
2015 VJIMC, 3
[b]Problem 3[/b]
Let $ P(x) = x^{2015} -2x^{2014}+1$ and $ Q(x) = x^{2015} -2x^{2014}-1$. Determine for each of the polynomials
$P$ and $Q$ whether it is a divisor of some nonzero polynomial $c_0 + c_{1}x +\ldots + c_{n}x^n$
n whose coefficients $c_i$ are all in the set $ \{ -1, 1\}$.
1955 Miklós Schweitzer, 8
[b]8.[/b] Show that on any tetrahedron there can be found three acute bihedral angles such that the faces including these angles count among them all faces of tetrahedron. [b](G. 10)[/b]
2011 IMC, 3
Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right)$.
2018 IMC, 1
Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent:
[list=1]
[*]There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;[/*]
[*]$\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.[/*]
[/list]
[i]Proposed by Tomáš Bárta, Charles University, Prague[/i]
2022 VTRMC, 4
Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.
1971 Putnam, B6
Let $\delta (x)$ be the greatest odd divisor of the positive integer $x$. Show that $| \sum_{n=1}^x \delta (n)/n -2x/3| <1,$ for all positive integers $x.$
2014 Contests, 3
Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.
KoMaL A Problems 2017/2018, A. 723
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that the limit
$$g(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-2f(x)+f(x-h)}{h^2}}$$
exists for all real $x$. Prove that $g(x)$ is constant if and only if $f(x)$ is a polynomial function whose degree is at most $2$.
1994 Putnam, 1
Suppose that a sequence $\{a_n\}_{n\ge 1}$ satisfies $0 < a_n \le a_{2n} + a_{2n+1}$ for all $n\in \mathbb{N}$. Prove that the series$\sum_{n=1}^{\infty} a_n$ diverges.
2015 VJIMC, 1
[b]Problem 1[/b]
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable on $\mathbb{R}$. Prove that there exists $x \in [0, 1]$ such that
$$\frac{4}{\pi} ( f(1) - f(0) ) = (1+x^2) f'(x) \ .$$
2019 Simon Marais Mathematical Competition, A2
Consider the operation $\ast$ that takes pair of integers and returns an integer according to the rule
$$a\ast b=a\times (b+1).$$
[list=a]
[*]For each positive integer $n$, determine all permutations $a_1,a_2,\dotsc , a_n$ of the set $\{ 1,2,\dotsc ,n\}$ that maximise the value of $$(\cdots ((a_1\ast a_2)\ast a_3) \ast \cdots \ast a_{n-1})\ast a_n.$$[/*]
[*]For each positive integer $n$, determine all permutations $b_1,b_2,\dotsc , b_n$ of the set $\{ 1,2,\dotsc ,n\}$ that maximise the value of $$b_1\ast (b_2\ast (b_3\ast \cdots \ast (b_{n-1}\ast b_n)\cdots )).$$[/*]
[/list]