Found problems: 876
1979 Putnam, A5
Denote by $\lceil x \rceil$ the greatest integer less than or equal to $x$ and by $S(x)$ the sequence $\lceil x \rceil, \lceil 2x \rceil, \lceil 3x \rceil, \dots.$ Prove that there are distinct real solutions $\alpha$ and $\beta$ of the equation $$x^3-10x^2+29x-25=0$$ such that infinitely many positive integers appear both in $S(\alpha)$ and in $S(\beta).$
1966 Putnam, B1
Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$.
2009 Miklós Schweitzer, 8
Let $ \{A_n\}_{n \in \mathbb{N}}$ be a sequence of measurable subsets of the real line which covers almost every point infinitely often. Prove, that there exists a set $ B \subset \mathbb{N}$ of zero density, such that $ \{A_n\}_{n \in B}$ also covers almost every point infinitely often. (The set $ B \subset \mathbb{N}$ is of zero density if $ \lim_{n \to \infty} \frac {\#\{B \cap \{0, \dots, n \minus{} 1\}\}}{n} \equal{} 0$.)
MIPT student olimpiad autumn 2022, 1
Prove that if a function $f:R \to R$ is bounded and its graph is closed as
subset of the $R^2$ plane, then the function f is continuous.
1998 Putnam, 4
Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2A_1=10$, $A_4=A_3A_2=101$, $A_5=A_4A_3=10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$.
2022 VTRMC, 2
Let $A$ and $B$ be the two foci of an ellipse and let $P$ be a point on this ellipse. Prove that the focal radii of $P$ (that is, the segments $\overline{AP}$ and $\overline{BP}$ ) form equal angles with the tangent to the ellipse at $P$.
2022 District Olympiad, P4
Let $I\subseteq \mathbb{R}$ be an open interval and $f:I\to\mathbb{R}$ a strictly monotonous function. Prove that for all $c\in I$ there exist $a,b\in I$ such that $c\in (a,b)$ and \[\int_a^bf(x) \ dx=f(c)\cdot (b-a).\]
2018 VTRMC, 6
For $n \in \mathbb{N}$, define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$. Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$.
2016 Miklós Schweitzer, 9
For $p_0,\dots,p_d\in\mathbb{R}^d$, let
\[
S(p_0,\dots,p_d)=\left\{ \alpha_0p_0+\dots+\alpha_dp_d : \alpha_i\le 1, \sum_{i=0}^d \alpha_i =1 \right\}.
\]
Let $\pi$ be an arbitrary probability distribution on $\mathbb{R}^d$, and choose $p_0,\dots,p_d$ independently with distribution $\pi$. Prove that the expectation of $\pi(S(p_0,\dots,p_d))$ is at least $1/(d+2)$.
2017 Korea USCM, 6
Given a positive integer $n$ and a real valued $n\times n$ matrix $A$. $J$ is $n\times n$ matrix with every entry $1$. Suppose $A$ satisfies the following relations.
$$A+A^T = \frac{1}{n} J, \quad AJ = \frac{1}{2} J$$
Show that $A^m-I$ is an invertible matrix for all positive odd integer $m$.
1990 Putnam, B5
Is there an infinite sequence $ a_0, a_1, a_2, \cdots $ of nonzero real numbers such that for $ n = 1, 2, 3, \cdots $ the polynomial \[ p_n(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \] has exactly $n$ distinct real roots?
2017 Brazil Undergrad MO, 1
A polynomial is called positivist if it can be written as a product of two non-constant polynomials with non-negative real coefficients. Let $f(x)$ be a polynomial of degree greater than one such that $f(x^n)$ is positivist for some positive integer $n$. Show that $f(x)$ is positivist.
1958 Miklós Schweitzer, 5
[b]5.[/b] Prove that neither the closed nor the open interval can be decomposed into finitely many mutually disjoint proper subsets which are all congruent by translation. [b](St. 2)[/b]
2018 Korea USCM, 3
$\Phi$ is a function defined on collection of bounded measurable subsets of $\mathbb{R}$ defined as
$$\Phi(S) = \iint_S (1-5x^2 + 4xy-5y^2 ) dx dy$$
Find the maximum value of $\Phi$.
2022 Miklós Schweitzer, 2
Original in Hungarian; translated with Google translate; polished by myself.
Let $n$ be a positive integer. Suppose that the sum of the matrices $A_1, \dots, A_n\in \Bbb R^{n\times n}$ is the identity matrix, but
$\sum\nolimits_{i = 1}^n\alpha_i A_i$ is singular whenever at least one of the coefficients $\alpha_i \in \Bbb R$ is zero.
a) Show that $\sum\nolimits_{i = 1}^n\alpha_i A_i$ is nonsingular if $\alpha_i\neq 0$ for all $i$.
b) Show that if the matrices $A_i$ are symmetric, then all of them have rank $1$.
1984 Miklós Schweitzer, 10
[b]10.[/b] Let $X_1, X_2, \dots $ be independent random variables with the same distribution
$P(X_i = 1) = P(X_i = -1)=\frac{1}{2}\qquad (i= 1, 2, \dots )$
Define
$S_0=0, Sn=X_1 +X_2+\dots +X_n \qquad (n=1, 2, \dots$ ),
$\xi (x,n) = \left | \{k : 0 \leq k \leq n, S_k= x \} \right |\qquad (x=0, \pm 1, \pm 2, \dots $),
and
$\alpha(n)= \left | \{ x: \xi(x,n)=a \} \right |\qquad (n=0,1,\dots$).
Prove that
$P(\lim \inf \alpha(n)=0) =1$
and that there is a number $0<c<\infty$ such that $P(\lim \inf \alpha(n)/\log n=c) =1$
([b]P.24[/b])
[P. Révész]
1966 Putnam, A1
Let $f(n)$ be the sum of the first $n$ terms of the sequence $0,1,1,2,2,3,3,4, \dots,$ where the $n$th term is given by $$a_n= \begin{cases} n/2 & \text{if } n \text{ is even,} \\ (n-1)/2 & \text{if } n \text{ is odd.} \end{cases}$$ Show that if $x$ and $y$ are positive integers and $x>y$ then $xy=f(x+y)-f(x-y)$.
MIPT student olimpiad autumn 2022, 2
Let $n \geq 3$ be an integer. Find the minimum degree of one algebraic (polynomial) equation that defines the set of vertices of the correct $n$-gon on plane $R^2$.
ICMC 5, 2
Evaluate
\[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\]
[i]Proposed by Ethan Tan[/i]
1958 Miklós Schweitzer, 11
[b]11.[/b] Let $a_n = (-1)^n (n= 1, 2, \dots , 2N)$. Denote by $A_{N}(x)$ the number of the sequences $1 \leq i_1 < i_2< \dots <i_N \leq 2N$ such that $a_{i_1}+a_{i_2}+ \dots +a_{i_N}< x \sqrt{\frac{N}{2}} (-\infty < x < \infty)$. Show that
$\lim_{N \to \infty} \frac{A_{N}(x)}{\binom{2N}{N}} = \frac {1}{\sqrt {2\pi}} \int_{-\infty}^{\infty} e^{-\frac{u^2}{2}} du$.
[b](N. 16)[/b]
1961 Miklós Schweitzer, 4
[b]4.[/b] Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1 $. Set
$ c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx $
and construct the following matrices of order $n$:
$ T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1} $
where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$:
$
S= \begin{bmatrix}
E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\
T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\
T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\
\dots & \dots & \dots & \dots & \dots & \dots \\
T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E
\end{bmatrix} $
($S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$).
Show that for any pair $(m , n) $ of positive integers, $S$ has only non-negative real eigenvalues. [b](R. 19)[/b]
2012 Putnam, 3
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that
(i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$
(ii) $ f(0)=1,$ and
(iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
Prove that $f$ is unique, and express $f(x)$ in closed form.
2018 Miklós Schweitzer, 10
In 3-dimensional hyperbolic space, we are given a plane $P$ and four distinct straight lines: the lines $a_1$ and $a_2$ are perpendicular to $P$; while the lines $r_1$ and $r_2$ do not intersect $P$, and their distances from $P$ are equal. Denote by $S_i$ the surface of revolution obtained by rotating $r_i$ around $a_i$. Show that the common points of $S_1$ and $S_2$ can be covered by two planes.
1989 Putnam, A2
Evaluate $\int^{a}_{0}{\int^{b}_{0}{e^{max(b^{2}x^{2},a^{2}y^{2})}dy dx}}$
2006 IMC, 6
The scores of this problem were:
one time 17/20 (by the runner-up)
one time 4/20 (by Andrei Negut)
one time 1/20 (by the winner)
the rest had zero... just to give an idea of the difficulty.
Let $A_{i},B_{i},S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that [list][*]not all $A_{i}$ have a common real eigenvector, [*]$A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, [*]$A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$.[/list] Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.