Found problems: 876
2019 IMC, 2
A four-digit number $YEAR$ is called [i]very good[/i] if the system
\begin{align*}
Yx+Ey+Az+Rw& =Y\\
Rx+Yy+Ez+Aw & = E\\\
Ax+Ry+Yz+Ew & = A\\
Ex+Ay+Rz+Yw &= R
\end{align*}
of linear equations in the variables $x,y,z$ and $w$ has at least two solutions. Find all very good $YEAR$s in the 21st century.
(The $21$st century starts in $2001$ and ends in $2100$.)
[i]Proposed by Tomáš Bárta, Charles University, Prague[/i]
1958 Miklós Schweitzer, 10
[b]10.[/b] Prove that the function
$f(x)= \int_{-\infty}^{\infty} \left (\frac{\sin\theta}{\theta} \right )^{2k}\cos (2x\theta) d\theta$
where $k$ is a positive integer, satisfies the following conditions:
[b](i)[/b] $f(x)=0$ if $\mid x \mid \geq k$ and $f(x) \geq 0$ elsewhere;
[b](ii)[/b] in interval $(l,l+1)$ $(l= -k, -k+1, \dots , k-1)$ the function $f(x)$ is a polynomial of degree $2k-1$ at most. [b](R. 7)[/b]
1954 Miklós Schweitzer, 3
[b]3.[/b] Is there a real-valued function $Af$, defined on the space of the functions, continuous on $[0,1]$, such that $f(x)\leq g(x) $ and$f(x)\not\equiv g(x) $ inply $Af< Ag$? Is this also true if the functions $f(x)$ are required to be monotonically increasing (rather than continuous) on $[0,1]$? [b](R.4)[/b]
2016 SEEMOUS, Problem 2
SEEMOUS 2016 COMPETITION PROBLEMS
2003 IMC, 4
Determine the set of all pairs (a,b) of positive integers for which the set of positive integers can be decomposed into 2 sets A and B so that $a\cdot A=b\cdot B$.
1958 Miklós Schweitzer, 1
[b]1.[/b] Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) [b](A. 14)[/b]
1995 Putnam, 5
A game starts with four heaps of beans, containing 3, 4, 5 and 6 beans. The two players move alternately. A move consists of taking [list]
(a) $\text{either}$ one bean from a heap, provided at least two beans are left behind in that heap,
(b) $\text{or}$ a complete heap of two or three beans.[/list]
The player who takes the last heap wins. To win the game, do you want to move first or second? Give a winning strategy.
2022 Miklós Schweitzer, 10
Is there a continuous function $f : \mathbb R \backslash \mathbb Q \to \mathbb R \backslash \mathbb Q$ for which the archetype of every irrational number has a positive Hausdorff dimension?
1961 Miklós Schweitzer, 2
[b]2.[/b] Show that a ring $R$ has a unit element if and only if any $R$-module $G$ can be written as a direct sum of $RG$ and of the trivial submodule of $G$. (An $R$-module is a linear space with $R$ as its scalar domain. $RG$ denotes the submodule generated by the elements of the form $rg$($r \in R, g \in G$). The trivial submodule of $G$ consists of the elements $g$ of $G$ for which $rg=0$ holds for every $r \in R$.) [b](A. 20)[/b]
2018 Korea USCM, 5
A real symmetric $2018\times 2018$ matrix $A=(a_{ij})$ satisfies $|a_{ij}-2018|\leq 1$ for every $1\leq i,j\leq 2018$. Denote the largest eigenvalue of $A$ by $\lambda(A)$. Find maximum and minumum value of $\lambda(A)$.
1979 Putnam, A4
Let $A$ be a set of $2n$ points in the plane, no three of which are collinear. Suppose that $n$ of them are colored red and the remaining $n$ blue. Prove or disprove: there are $n$ closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of $A$ having different colors.
2017 VJIMC, 4
Let $f:(1,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $f(x) \le x^2 \log(x)$ and $f'(x)>0$ for every $x \in (1,\infty)$. Prove that
\[\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.\]
MIPT student olimpiad autumn 2024, 1
$F$* is the multiplicative group of the field $F$.
$F$* is of finitely generated.
Is it true that $F$* is cyclic?
Additional question: (wasn’t at the olympiad)
$K$* is the multiplicative group of the field $K$.
$L \subseteq $$K$* is a finitely generated subgroup.
Is it true that $L$ is cyclic?
1997 Putnam, 3
Evaluate the following :
\[ \int_{0}^{\infty}\left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots \right)\;\left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2\cdot 6^2}+\cdots \right)\,\mathrm{d}x \]
1976 Putnam, 3
Suppose that we have $n$ events $A_1,\dots, A_n,$ each of which has probability at least $1-a$ of occuring, where $a<1/4.$ Further suppose that $A_i$ and $A_j$ are mutually independent if $|i-j|>1.$ Assume as known that the recurrence $u_{k+1}=u_k-au_{k-1}, u_0=1, u_1=1-a,$ defines positive real numb $u_k$ for $k=0,1,\dots.$ Show that the probability of all of $A_1,\dots, A_n$ occuring is at least $u_n.$
1999 Putnam, 1
Right triangle $ABC$ has right angle at $C$ and $\angle BAC=\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE=\theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\to 0}|EF|$.
2016 SEEMOUS, Problem 1
SEEMOUS 2016 COMPETITION PROBLEMS
2000 VJIMC, Problem 1
Is there a countable set $Y$ and an uncountable family $\mathcal F$ of its subsets such that for every two distinct $A,B\in\mathcal F$, their intersection $A\cap B$ is finite?
2019 SEEMOUS, 2
Let $A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})$. Prove that there exist $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}$ such that:
$$\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2) $$
1958 Miklós Schweitzer, 7
[b]7.[/b] Let $a_0$ and $a_1$ be arbitrary real numbers, and let
$a_{n+1}=a_n + \frac{2}{n+1}a_{n-1}$ $(n= 1, 2, \dots)$
Show that the sequence $\left (\frac{a_n}{n^2} \right )_{n=1}^{\infty}$ is convergent and find its limit. [b](S. 10)[/b]
2016 Korea USCM, 5
For $f(x) = \cos\left(\frac{3\sqrt{3}\pi}{8}(x-x^3 ) \right)$, find the value of
$$\lim_{t\to\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} + \lim_{t\to-\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} $$
2001 Miklós Schweitzer, 6
Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the
$$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$
inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is
$$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$
2013 Putnam, 2
Let $S$ be the set of all positive integers that are [i]not[/i] perfect squares. For $n$ in $S,$ consider choices of integers $a_1,a_2,\dots, a_r$ such that $n<a_1<a_2<\cdots<a_r$ and $n\cdot a_1\cdot a_2\cdots a_r$ is a perfect square, and let $f(n)$ be the minimum of $a_r$ over all such choices. For example, $2\cdot 3\cdot 6$ is a perfect square, while $2\cdot 3,2\cdot 4, 2\cdot 5, 2\cdot 3\cdot 4,$ $2\cdot 3\cdot 5, 2\cdot 4\cdot 5,$ and $2\cdot 3\cdot 4\cdot 5$ are not, and so $f(2)=6.$ Show that the function $f$ from $S$ to the integers is one-to-one.
1990 Putnam, A2
Is $ \sqrt{2} $ the limit of a sequence of numbers of the form $ \sqrt[3]{n} - \sqrt[3]{m} $, where $ n, m = 0, 1, 2, \cdots $.
2004 Putnam, A5
An $m\times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $\frac12.$ we say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q,$ in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than $\frac{mn}8.$