Found problems: 17
Let $m,n\geq2$ be positive integers, and let $a_1,a_2,\ldots ,a_n$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_1,e_2,\ldots,e_n$, not all zero, with $\left|{\,e}_i\,\right|<m$ for all $i$, such that $e_1a_1+e_2a_2+\,\ldots\,+e_na_n$ is a multiple of $m^n$.
Let $n$ be an odd positive integer greater than $2$, and consider a regular $n$-gon $\mathcal{G}$ in the plane centered at the origin. Let a [i]subpolygon[/i] $\mathcal{G}'$ be a polygon with at least $3$ vertices whose vertex set is a subset of that of $\mathcal{G}$. Say $\mathcal{G}'$ is [i]well-centered[/i] if its centroid is the origin. Also, say $\mathcal{G}'$ is [i]decomposable[/i] if its vertex set can be written as the disjoint union of regular polygons with at least $3$ vertices. Show that all well-centered subpolygons are decomposable if and only if $n$ has at most two distinct prime divisors.
Show that there are no positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ such that
$$(1+a_1 \omega)(1+a_2 \omega)(1+a_3 \omega)(1+a_4 \omega)(1+a_5 \omega)(1+a_6 \omega)$$
is an integer where $\omega$ is an imaginary $5$th root of unity.
Let $q=p^r$ for a prime number $p$ and positive integer $r$. Let $\zeta = e^{\frac{2\pi i}{q}}$. Find the least positive integer $n$ such that
\[\sum_{\substack{1\leq k\leq q\\ \gcd(k,p)=1}} \frac{1}{(1-\zeta^k)^n}\]
is not an integer. (The sum is over all $1\leq k\leq q$ with $p$ not dividing $k$.)
[i]Victor Wang[/i]
Let $p$ and $q$ be prime numbers with $p<q$. Suppose that in a convex polygon $P_1,P_2,…,P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that
$$P_1P_2+P_2P_3+\cdots +P_kP_{k+1}\geqslant \frac{k^3+k}{2}$$holds for every integer $k$ with $1\leqslant k\leqslant p$.
[i]Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin[/i]
$p(x)$ is a polynomial of degree $n$ with every coefficient $0 $ or $\pm1$, and $p(x)$ is divisible by $ (x - 1)^k$ for some integer $ k > 0$. $q$ is a prime such that $\frac{q}{\ln q}< \frac{k}{\ln n+1}$. Show that the complex $q$-th roots of unity must be roots of $ p(x). $
Let $\omega = e^{2\pi i /5}$ be a primitive fifth root of unity. Prove that there do not exist integers $a, b, c, d, k$ with $k > 1$ such that \[(a + b \omega + c \omega^2 + d \omega^3)^{k}=1+\omega.\]
[i]Carl Lian[/i]
Find integers \(a\) and \(b\) such that
\[a^b=3^0\binom{2020}{0}-3^1\binom{2020}{2}+3^2\binom{2020}{4}-\cdots+3^{1010}\binom{2020}{2020}.\]
[i]proposed by the ICMC Problem Committee[/i]
Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?
Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let
$$b_k=\sum_{i=1}^n a_i \omega^{ki}$$
for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.
Compute \[\log_2\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{2\pi iab/2015}\right)\right)\] Here $i$ is the imaginary unit (that is, $i^2=-1$).
(a) For given complex numbers $a_1,a_2,a_3,a_4$, we define a function $P:\mathbb C\to\mathbb C$ by $P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z$. Let $w_k=e^{2ki\pi/5}$, where $k=0,\ldots,4$. Prove that
$$P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.$$(b) Let $A_1,A_2,A_3,A_4,A_5$ be five points in the plane. A pentagon is inscribed in the circle with center $A_1$ and radius $R$. Prove that there is a vertex $S$ of the pentagon for which
$$SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.$$
Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?
Given a finite set $X$, two positive integers $n,k$, and a map $f:X\to X$. Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$,$i=1,2,3,\ldots$. It is known that for any $x\in X$,$f^{(n)}(x)=x$.
Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$.
Prove that:
(1)$\frac{1}n \sum_{j=1}^n m_j\sin {\frac{2kj\pi}{n}}=0$
(2)$\frac{1}n \sum_{j=1}^n m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.
Let $m>1$ be an odd integer. Let $n=2m$ and $\theta=e^{2\pi i\slash n}$. Find integers $a_{1},\ldots,a_{k}$ such that
$\sum_{i=1}^{k}a_{i}\theta^{i}=\frac{1}{1-\theta}$.
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.
Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let
$$b_k=\sum_{i=1}^n a_i \omega^{ki}$$
for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.