Found problems: 876
1977 Putnam, A3
Let $u,f$ and $g$ be functions, defined for all real numbers $x$, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text{ and } \frac{u(x+4)+u(x-4)}{2}=g(x).$$ Determine $u(x)$ in terms of $f$ and $g$.
2020 Putnam, A3
Let $a_0=\pi /2$, and let $a_n=\sin (a_{n-1})$ for $n\ge 1$. Determine whether
\[ \sum_{n=1}^{\infty}a_n^2 \]
converges.
2007 IMC, 5
Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.
2021 Alibaba Global Math Competition, 9
Let $\varepsilon$ be positive constant and $u$ satisfies that
\[
\begin{cases} (\partial_t-\varepsilon\partial_x^2-\partial_y^2)u=0, & (t,x,y) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_+,\\ \partial_y u\vert_{y=0}=\partial_x h, &\\u\vert_{t=0}=0. & \end{cases}\]
Here $h(t,x)$ is a smooth Schwartz function. Define the operator $e^{a\langle D\rangle}$
\[\mathcal{F}_x(e^{a\langle D\rangle} f)(k)=e^{a\langle k\rangle} \mathcal{F}_x(f)(k), \quad \langle k\rangle=1+\vert k\vert,\]
where $\mathcal{F}_x$ stands for the Fourier transform in $x$. Show that
\[\int_0^T \|e^{(1-s)\langle D\rangle} u\|_{L_{x,y}^2}^2 ds \le C \int_0^T \|e^{(1-s)\langle D\rangle} h\|_{H_x^{\frac{1}{4}}}^2 ds\]
with constant $C$ independent of $\varepsilon, T$ and $h$.
1953 Miklós Schweitzer, 8
[b]8.[/b] Does there exist a Euclidean ring which is properly contained in the field $V$ of real numbers, and whose quotient field is $V$? [b](A.21)[/b]
2001 Putnam, 1
Consider a set $S$ and a binary operation $*$, i.e. for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b \in S$.
2023 IMC, 10
For every positive integer $n$, let $f(n)$, $g(n)$ be the minimal positive integers such that
\[1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}.\]
Determine whether there exists a positive integer $n$ for which $g(n)>n^{0.999n}$.
2003 Miklós Schweitzer, 4
Let $\{a_{n,1},\ldots, a_{n,n} \}_{n=1}^{\infty}$ integers such that $a_{n,i}\neq a_{n,j}$ for $1\le i<j\le n\, , n=2,3,\ldots$ and let $\left\langle y\right\rangle\in [0,1)$ denote the fractional part of the real number $y$. Show that there exists a real sequence $\{ x_n\}_{n=1}^{\infty}$ such that the numbers $\langle a_{n,1}x_n \rangle, \ldots, \langle a_{n,n}x_n \rangle$ are asymptotically uniformly distributed on the interval $[0,1]$.
(translated by L. Erdős)
2022 Miklós Schweitzer, 9
Plane vectors form a group for addition. Show that this group has a generator system of every set $S$ that contains a Borel subset of positive linear measure of a circular arc.
MIPT student olimpiad spring 2023, 4
Is it true that if two linear subspaces $V$ and $W$ of a Hilbert space are closed, then their sum $V+W$ is also closed?
2001 Putnam, 6
Can an arc of a parabola inside a circle of radius $1$ have a length greater than $4$?
1954 Miklós Schweitzer, 9
[b]9.[/b] Lep $p$ be a connected non-closed broken line without self-intersection in the plane $\varphi $. Prove that if $v$ is a non-zero vector in $\varphi $ and $p$ has a commom point with the broken line $p+v$, then $p$ has a common point with the broken line $p+\alpha v$ too, where $\alpha =\frac{1}{n}$ and $n$ is a positive integer. Does a similar statemente hold for other positive values of $\alpha$? ($p+v$ denotes the broken line obtained from $p$ through displacemente by the vector $v$.) [b](G. 1)[/b]
2000 Miklós Schweitzer, 3
Prove that for every integer $n\ge 3$ there exists $N(n)$ with the following property: whenever $P$ is a set of at least $N(n)$ points of the plane such that any three points of $P$ determines a nondegenerate triangle containing at most one point of $P$ in its interior, then $P$ contains the vertices of a convex $n$-gon whose interior does not contain any point of $P$.
1994 Putnam, 2
Let $A$ be the area of the region in the first quadrant bounded by the line $y = \frac{x}{2}$, the x-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx,$ the y-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1.$
1991 Putnam, A5
A5) Find the maximum value of $\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx$ for $0\leq y\leq 1$.
I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted $f(y)$, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule[/url] we have
$\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}$
Now what?
ICMC 4, 2
Let $p > 3$ be a prime number. A sequence of $p-1$ integers $a_1,a_2, \dots, a_{p-1}$ is called [i]wonky[/i] if they are distinct modulo \(p\) and $a_ia_{i+2} \not\equiv a_{i+1}^2 \pmod p$ for all \(i \in \{1, 2, \dots, p-1\}\), where \(a_p = a_1\) and \(a_{p+1} = a_2\). Does there always exist a wonky sequence such that $$a_1a_2, \qquad a_1a_2+a_2a_3, \qquad \dots, \qquad a_1a_2+\cdots +a_{p-1}a_1,$$ are all distinct modulo $p$?
[i]Proposed by Harun Khan[/i]
1986 Miklós Schweitzer, 2
Show that if $k\leq \frac n2$ and $\mathcal F$ is a family $k\times k$ submatrices of an $n\times n$ matrix such that any two intersect then
$$|\mathcal F|\leq \binom{n-1}{k-1}^2$$
[Gy. Katona]
1979 Putnam, A2
Establish necessary and sufficient conditions on the constant $k$ for the existence of a continuous real valued function $f(x)$ satisfying $$f(f(x))=kx^9$$ for all real $x$.
2018 Miklós Schweitzer, 9
Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function, and suppose that the sequence $f^{(n)}$ of derivatives converges pointwise. Prove that $f^{(n)}(z)\to Ce^z$ pointwise for a suitable complex number $C$.
ICMC 8, 1
Joe the Jaguar is on an infinite grid of unit squares, starting at the centre of one of them. At the $k$-th minute, Joe must jump a distance of $k$ units in any direction. For which $n$ is it possible for Joe to be inside or on the edge of the starting square after $n$ minutes?
1949 Miklós Schweitzer, 4
Let $ A$ and $ B$ be two disjoint sets in the interval $ (0,1)$ . Denoting by $ \mu$ the Lebesgue measure on the real line, let $ \mu(A)>0$ and $ \mu(B)>0$ . Let further $ n$ be a positive integer and $ \lambda \equal{}\frac1n$ . Show that there exists a subinterval $ (c,d)$ of $ (0,1)$ for which $ \mu(A\cap (c,d))\equal{}\lambda \mu(A)$ and $ \mu(B\cap (c,d))\equal{}\lambda \mu(B)$ . Show further that this is not true if $ \lambda$ is not of the form $ \frac1n$.
1997 Putnam, 2
Players $1,2,\ldots n$ are seated around a table, and each has a single penny. Player $1$ passes a penny to Player $2$, who then passes two pennies to Player $3$, who then passes one penny to player $4$, who then passes two pennies to Player $5$ and so on, players alternately pass one or two pennies to the next player who still has some pennies. The player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers $n$ for which some player ends up with all the $n$ pennies.
1984 Miklós Schweitzer, 4
[b]4.[/b] Let $x_1 , x_2 , y_1 , y_2 , z_1 , z_2 $ be transcendental numbers. Suppose that any 3 of them are algebraically independent, and among the 15 four-tuples on $\{x_1 , x_2 , y_1, y_2 \}$, $\{ x_1 , x_2 , z_1 , z_2 \} $ and $ \{y_1 , y_2 , z_1 , z_2 \} $ are algebraically dependent. Prove that there exists a transcendental number $t$ that depends algebraically on each of the pairs $\{ x_1 , x_2\}$ , $\{ y_1 , y_2 \}$, and $\{ z_1 , z_2 \}$. ([b]A.37[/b])
[L. Lovász]
2016 SEEMOUS, Problem 3
SEEMOUS 2016 COMPETITION PROBLEMS
1956 Miklós Schweitzer, 6
[b]6.[/b] Show that the number of the faces of a convex polyhedron is even if every face is centrally simmetric. [b](G. 12)[/b]