Found problems: 876
2021 Alibaba Global Math Competition, 19
Find all real numbers of the form $\sqrt[p]{2021+\sqrt[q]{a}}$ that can be expressed as a linear combination of roots of unity with rational coefficients, where $p$ and $q$ are (possible the same) prime numbers, and $a>1$ is an integer, which is not a $q$-th power.
1977 Putnam, B4
Let $C$ be a continuous closed curve in the plane which does not cross itself and let $Q$ be a point inside $C$. Show that there exists points $P_1$ and $P_2$ on $C$ such that $Q$ is the midpoint of the line segment $P_1P_2.$
2005 IMC, 1
1. Let $f(x)=x^2+bx+c$, M = {x | |f(x)|<1}. Prove $|M|\leq 2\sqrt{2}$ (|...| = length of interval(s))
2017 IMC, 8
Define the sequence $A_1,A_2,\ldots$ of matrices by the following recurrence: $$ A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \quad A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \\ \end{pmatrix} \quad (n=1,2,\ldots) $$ where $I_m$ is the $m\times m$ identity matrix.
Prove that $A_n$ has $n+1$ distinct integer eigenvalues $\lambda_0< \lambda_1<\ldots <\lambda_n$ with multiplicities $\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}$, respectively.
1996 Putnam, 1
Define a $\emph{selfish}$ set to be a set which has its own cardinality as its element. And a set is a $\emph{minimal }\text{ selfish}$ set if none of its proper subsets are $\emph{selfish}$. Find with proof the number of $\text{minimal selfish}$ subsets of $\{1,2,\cdots ,n\}$.
2022 Miklós Schweitzer, 8
Original in Hungarian; translated with Google translate; polished by myself.
Prove that, the signs $\varepsilon_n = \pm 1$ can be chosen such that the function $f(s) = \sum_{n = 1}^\infty\frac{\varepsilon_n}{n^s}\colon \{s\in\Bbb C:\operatorname{Re}s > 1\}\to \Bbb C$ converges to every complex value at every point $\xi \in \{s\in\Bbb C:\operatorname{Re}s = 1\}$ (i.e. for every $\xi \in \{s\in\Bbb C:\operatorname{Re}s = 1\}$ and every $z \in \Bbb C$, there exists a sequence $s_n \to \xi$, $\operatorname{Re}s_n > 1$, for which $f(s_n) \to z$).
2015 VJIMC, 1
[b]Problem 1 [/b]
Let $A$ and $B$ be two $3 \times 3$ matrices with real entries. Prove that
$$ A-(A^{-1} +(B^{-1}-A)^{-1})^{-1} =ABA\ , $$
provided all the inverses appearing on the left-hand side of the equality exist.
2018 VTRMC, 2
Let $A, B \in M_6 (\mathbb{Z} )$ such that $A \equiv I \equiv B \text{ mod }3$ and $A^3 B^3 A^3 = B^3$. Prove that $A = I$. Here $M_6 (\mathbb{Z} )$ indicates the $6$ by $6$ matrices with integer entries, $I$ is the identity matrix, and $X \equiv Y \text{ mod }3$ means all entries of $X-Y$ are divisible by $3$.
2000 Miklós Schweitzer, 5
Prove that for every $\varepsilon >0$ there exists a positive integer $n$ and there are positive numbers $a_1, \ldots, a_n$ such that for every $\varepsilon < x < 2\pi - \varepsilon$ we have
$$\sum_{k=1}^n a_k\cos kx < -\frac{1}{\varepsilon}\left| \sum_{k=1}^n a_k\sin kx\right|$$.
ICMC 2, 2
This question, again, comprises two independent parts.
(i) Show that if \((k+1)\) integers are chosen from \(\left\{1,2,3,...,2k+1\right\}\), then among the chosen integers there are always two that are coprime.
(ii) Let \(A=\left\{1,2,\ldots,n\right\}.\) Prove that if \(n>11\) then there is a bijective map \(f: A\to A\) with the property that, for every \(a\in A\), exactly one of \(f(f(f(f(a))))=a\) and \(f(f(f(f(f(a)))))=a\) holds.
1998 VJIMC, Problem 3
Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.
2014 IMC, 5
For every positive integer $n$, denote by $D_n$ the number of permutations $(x_1, \dots, x_n)$ of $(1,2,\dots, n)$ such that $x_j\neq j$ for every $1\le j\le n$. For $1\le k\le \frac{n}{2}$, denote by $\Delta (n,k)$ the number of permutations $(x_1,\dots, x_n)$ of $(1,2,\dots, n)$ such that $x_i=k+i$ for every $1\le i\le k$ and $x_j\neq j$ for every $1\le j\le n$. Prove that
$$\Delta (n,k)=\sum_{i=0}^{k=1} \binom{k-1}{i} \frac{D_{(n+1)-(k+i)}}{n-(k+i)}$$
(Proposed by Combinatorics; Ferdowsi University of Mashhad, Iran; Mirzavaziri)
1979 Putnam, A6
Let $0\leq p_i \leq 1$ for $i=1,2, \dots, n.$ Show that $$\sum_{i=1}^{n} \frac{1}{|x-p_i|} \leq 8n(1+1/3+1/5+\dots +\frac{1}{2n-1})$$ for some $x$ satisfying $0\leq x \leq 1.$
1997 Putnam, 1
For all reals $x$ define $\{x\}$ to be the difference between $x$ and the closest integer to $x$. For each positive integer $n$ evaluate :
\[ S_n=\sum_{m=1}^{6n-1}\min \left(\left\{\frac{m}{6n}\right\},\left\{\frac{m}{3n}\right\}\right) \]
1961 Miklós Schweitzer, 3
[b]3.[/b] Let $f(x)= x^n +a_1 x^(n-1)+ \dots + a_n$ ($n\geq 1$) be an irreducible polynomial over the field $K$. Show that every non-zero matrix commuting with the matrix
$
\begin{bmatrix}
0 & 1 & 0 & \dots & 0 & 0 \\
0 & 0 & 1 & \dots & 0 & 0 \\
\dots & \dots & \dots & \dots & \dots & \dots \\
0 & 0 & 0 & \dots & 0 & 1 \\
-a_n & -a_{n-1} & -a_{n-2} & \dots & -a_2 & -a_1
\end{bmatrix} $
is invertible. [b](A. 4)[/b]
2004 IMC, 4
Suppose $n\geq 4$ and let $S$ be a finite set of points in the space ($\mathbb{R}^3$), no four of which lie in a plane. Assume that the points in $S$ can be colored with red and blue such that any sphere which intersects $S$ in at least 4 points has the property that exactly half of the points in the intersection of $S$ and the sphere are blue. Prove that all the points of $S$ lie on a sphere.
1998 Putnam, 6
Prove that, for any integers $a,b,c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.
2022 Miklós Schweitzer, 1
We say that a set $A \subset \mathbb Z$ is irregular if, for any different elements $x, y \in A$, there is no element of the form $x + k(y -x)$ different from $x$ and $y$ (where $k$ is an integer). Is there an infinite irregular set?
2004 Putnam, A6
Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that
$\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$
$\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$
2015 VJIMC, 2
[b]Problem 2[/b]
Determine all pairs $(n, m)$ of positive integers satisfying the equation
$$5^n = 6m^2 + 1\ . $$
2015 IMC, 2
For a positive integer $n$, let $f(n)$ be the number obtained by
writing $n$ in binary and replacing every 0 with 1 and vice
versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in
binary, therefore $f(23) =8$. Prove that
\[\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.\]
When does equality hold?
(Proposed by Stephan Wagner, Stellenbosch University)
2019 IMC, 8
Let $x_1,\ldots,x_n$ be real numbers. For any set $I\subset\{1,2,…,n\}$ let $s(I)=\sum_{i\in I}x_i$. Assume that the function $I\to s(I)$ takes on at least $1.8^n$ values where $I$ runs over all $2^n$ subsets of $\{1,2,…,n\}$. Prove that the number of sets $I\subset \{1,2,…,n\}$ for which $s(I)=2019$ does not exceed $1.7^n$.
[i]Proposed by Fedor Part and Fedor Petrov, St. Petersburg State University[/i]
1956 Miklós Schweitzer, 4
[b]4.[/b] Denoting by $a(n)$ the greatest prime factor of the positive integer $n$, show that
$S= \sum_{n=1}^{\infty } \frac{1}{na(n)}$
is convergente. [b](N. 13)[/b]
1996 Putnam, 4
$S$ be a set of ordered triples $(a,b,c)$ of distinct elements of a finite set $A$. Suppose that
[list=1]
[*] $(a,b,c)\in S\iff (b,c,a)\in S$
[*] $(a,b,c)\in S\iff (c,b,a)\not\in S$
[*] $(a,b,c),(c,d,a)\text{ both }\in S\iff (b,c,d),(d,a,b)\text{ both }\in S$[/list]
Prove there exists $g: A\to \mathbb{R}$, such that $g$ is one-one and $g(a)<g(b)<g(c)\implies (a,b,c)\in S$
2020 Simon Marais Mathematics Competition, A3
Determine the set of real numbers $\alpha$ that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\]
where $x_0,x_1,x_2,\dots$ is an increasing sequence of real numbers with $x_0=1$.