Found problems: 85335
For each positive integer $n$, let
\[a_n = \frac {(n + 9)!}{(n - 1)!}.\]
Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 7\qquad
\textbf{(E)}\ 9$
Let $p,q$ prime numbers such that $$p+q \mid p^3-q^3$$
Show that $p=q$.
Justin is being served two different types of chips, A-chips, and B-chips. If there are $3$ B-chips and $5$ A-chips, and if Justin randomly grabs $3$ chips, what is the probability that none of them are A-chips?
In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?
Let $n\geq 2$ be an integer and denote by $H_{n}$ the set of column vectors $^{T}(x_{1},\ x_{2},\ \ldots, x_{n})\in\mathbb{R}^{n}$, such that $\sum |x_{i}|=1$.
Prove that there exist only a finite number of matrices $A\in\mathcal{M}_{n}(\mathbb{R})$ such that the linear map $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $f(x)=Ax$ has the property $f(H_{n})=H_{n}$.
[hide="Comment"]In the contest, the problem was given with a) and b):
a) Prove the above for $n=2$;
b) Prove the above for $n\geq 3$ as well.[/hide]
Let $a,b$ be real numbers and $k$ a positive integer. Show that
$$ \left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1$$
whenever the left side is defined.
Let $n>1$ be an integer and let $f_1$, $f_2$, ..., $f_{n!}$ be the $n!$ permutations of $1$, $2$, ..., $n$. (Each $f_i$ is a bijective function from $\{1,2,...,n\}$ to itself.) For each permutation $f_i$, let us define $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$. Find $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$.
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 $1$.
Find all $x \in R$ such that $$ x - \left[ \frac{x}{2016} \right]= 2016$$, where $[k]$ represents the largest smallest integer or equal to $k$.
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.
[i]Proposed by Marcin Kuczma, Poland[/i]
[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers.
This number $k$ is called [i]weight[/i] of $n$.
[b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.
A $2010\times 2010$ board is divided into corner-shaped figures of three cells. Prove that it is possible to mark one cell in each figure such that each row and each column will have the same number of marked cells.
[i]I. Bogdanov & O. Podlipsky[/i]
The polynomial $f(x)=1-x+x^2-x^3+\cdots-x^{19}+x^{20}$ is written into the form $g(y)=a_0+a_1y+a_2y^2+\cdots+a_{20}y^{20}$, where $y=x-4$, then $a_0+a_1+\cdots+a_{20}=$________.
Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$
Show that
\[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \]
for all positive real numbers $a, \: b, \: c.$
The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $. Let $f(x)=\frac{e^x}{x}$.
Suppose $f$ is differentiable infinitely many times in $(0,\infty) $. Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$
$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \]
where $R$ is the radius of the circumscribing circle.
Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.
A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$
$2019$ circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?
Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.
A sequence of integers $a_1$, $a_2$, $a_3$, $\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?
Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.
Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$