Found problems: 876
1977 Putnam, A5
Prove that $$\binom{pa}{pb}=\binom{a}{b} (\text{mod } p)$$ for all integers $p,a,$ and $b$ with $p$ a prime, $p>0,$ and $a>b>0.$
2002 Miklós Schweitzer, 10
Let $X_1, X_2, \ldots$ be independent random variables of the same distribution such that their joint distribution is discrete and is concentrated on infinitely many different values. Let $a_n$ denote the probability that $X_1,\ldots, X_{n+1}$ are all different on the condition that $X_1,\ldots, X_n$ are all different ($n\ge 1$). Show that
(a) $a_n$ is strictly decreasing and tends to $0$ as $n\to \infty$; and
(b) for any sequence $1\le f(1)\le f(2) < \ldots$ of positive integers the joint distribution of $X_1, X_2, \ldots$ can be chosen such that
$$\limsup_{n\to\infty}\frac{a_{f(n)}}{a_n}=1$$
holds.
2009 IMC, 2
Let $A,B,C$ be real square matrices of the same order, and suppose $A$ is invertible. Prove that
\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]
1959 Miklós Schweitzer, 8
[b]8.[/b] An Oblique lattice-cubs is a lattice-cube of the three-dimensional fundamental lattice no edge of which is perpendicular to any coordinate axis. Prove that for any integer $h= 8n-1$ ($n= 1, 2, \dots $) there existis an oblique lattice-cube with edges of length $h$. Propose a method for finding such a cube. [b](N. 20)[/b]
1956 Miklós Schweitzer, 1
[b]1.[/b] Solve without use of determinants the following system of linear equations:
$\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k$ ($k= 0,1, \dots , n$),
where $\alpha$ is a fixed real number. [b](A. 7)[/b]
ICMC 4, 1
A set of points in the plane is called [i]sane[/i] if no three points are collinear and the angle between any three distinct points is a rational number of degrees.
(a) Does there exist a countably infinite sane set $\mathcal{P}$?
(b) Does there exist an uncountably infinite sane set $\mathcal{Q}$?
[i]Proposed by Tony Wang[/i]
2013 IMC, 4
Let $\displaystyle{n \geqslant 3}$ and let $\displaystyle{{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define $\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$. Prove that:
\[\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.\]
[i]Proposed by Géza Kós, Eötvös University, Budapest.[/i]
1984 Miklós Schweitzer, 2
[b]2.[/b] Show that threre exist a compact set $K \subset \mathbb{R}$ and a set $A \subset \mathbb{R}$ of type $F_{\sigma}$ such that the set
$\{ x\in \mathbb{R} : K+x \subset A\}$
is not Borel-measurable (here $K+x = \{y+x : y \in K\}$). ([b]M.16[/b])
[M. Laczkovich]
2014 Paenza, 5
Let $\mathbb{A}$ be the least subset of finite sequences of nonnegative integers that satisfies the following two properties:
-$(0,0) \in \mathbb{A}$
- If $(a_1,\ldots,a_n)\in \mathbb{A}$ then
$(a_1,\ldots,a_{i-2},a_{i-1}+1,1,a_{i}+1,a_{i+1},\ldots,a_n)\in \mathbb{A}$ for all $i\in \{2,\ldots,n\}$.
For each $n\geq 2$, let $\mathbb{B}(n)$ be the set of sequences in $\mathbb{A}$ with $n$ terms. Find the number of elements of $\mathbb{B}$.
1995 Putnam, 6
For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that
\[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \]
\[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]
1985 Miklós Schweitzer, 7
Let $p_1$ and $p_2$ be positive real numbers. Prove that there exist functions $f_i\colon \mathbb R \rightarrow \mathbb R$ such that the smallest positive period of $f_i$ is $p_i\, (i=1, 2)$, and $f_1-f_2$ is also periodic. [J. Riman]
2004 IMC, 3
Let $A_n$ be the set of all the sums $\displaystyle \sum_{k=1}^n \arcsin x_k $, where $n\geq 2$, $x_k \in [0,1]$, and $\displaystyle \sum^n_{k=1} x_k = 1$.
a) Prove that $A_n$ is an interval.
b) Let $a_n$ be the length of the interval $A_n$. Compute $\displaystyle \lim_{n\to \infty} a_n$.
2004 IMC, 2
Let $f_1(x)=x^2-1$, and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ have?
2018 Romania National Olympiad, 2
Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$
For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$
Determine $\min_{f \in \mathcal{F}}I(f).$
[i]Liviu Vlaicu[/i]
2012 Putnam, 1
Let $S$ be a class of functions from $[0,\infty)$ to $[0,\infty)$ that satisfies:
(i) The functions $f_1(x)=e^x-1$ and $f_2(x)=\ln(x+1)$ are in $S;$
(ii) If $f(x)$ and $g(x)$ are in $S,$ the functions $f(x)+g(x)$ and $f(g(x))$ are in $S;$
(iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x)\ge g(x)$ for all $x\ge 0,$ then the function $f(x)-g(x)$ is in $S.$
Prove that if $f(x)$ and $g(x)$ are in $S,$ then the function $f(x)g(x)$ is also in $S.$
2015 IMC, 6
Prove that
$$\sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} < 2.$$
Proposed by Ivan Krijan, University of Zagreb
2011 Putnam, B1
Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]
2019 IMC, 1
Evaluate the product
$$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.$$
[i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan[/i]
1971 Putnam, A6
Let $c$ be a real number such that $n^c$ is an integer for every positive integer $n$. Show that $c$ is a non-negative integer.
1971 Putnam, A5
A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points ($a$ and $b$ are positive integers with $a$ greater than $b$), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is $58$. Find $a$ and $b$.
2018 Miklós Schweitzer, 2
A family $\mathcal{F}$ of sets is called [i]really neat[/i] if for any $A,B\in \mathcal{F}$, there is a set $C\in \mathcal{F}$ such that $A\cup B = A\cup C=B\cup C$. Let
$$f(n)=\min \left\{ \max_{A\in \mathcal{F}} |A| \colon \mathcal{F} \text{ is really neat and } |\cup \mathcal{F}| =n\right\} .$$
Prove that the sequence $f(n)/n$ converges and find its limit.
2015 Miklos Schweitzer, 6
Let $G$ be the permutation group of a finite set $\Omega$.Consider $S\subset G$ such that $1\in S$ and for any $x,y\in \Omega$ there exists a unique element $\sigma \in S$ such that $\sigma (x)=y$.Prove that,if the elements of $S \setminus \{1\}$ are conjugate in $G$,then $G$ is $2-$transitive on $\Omega$
1995 Putnam, 3
The number $d_1d_2\cdots d_9$ has nine (not necessarily distinct) decimal digits. The number $e_1e_2\cdots e_9$ is such that each of the nine $9$-digit numbers formed by replacing just one of the digits $d_i$ in $d_1d_2\cdots d_9$ by the corresponding digit $e_i \;\;(1 \le i \le 9)$ is divisible by $7$. The number $f_1f_2\cdots f_9$ is related to $e_1e_2\cdots e_9$ is the same way: that is, each of the nine numbers formed by replacing one of the $e_i$ by the corresponding $f_i$ is divisible by $7$. Show that, for each $i$, $d_i-f_i$ is divisible by $7$. [For example, if $d_1d_2\cdots d_9 = 199501996$, then $e_6$ may be $2$ or $9$, since $199502996$ and $199509996$ are multiples of $7$.]
2011 Putnam, B2
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$
ICMC 6, 2
Show that if the distance between opposite edges of a tetrahedron is at least $1$, then its volume is at least $\frac{1}{3}$.
[i]Proposed by Simeon Kiflie[/i]