Found problems: 85335
Prove that for every natural number $n > 1$ there exists a suquence $a_1$,$a_2$, $...$, $a_n$ of the numbers $1,2,...,n$ such that for each $k \in \{1,2,...,n-1\}$ the number $a_{k+1}$ divides $a_1+a_2+...+a_k$.
Let $\alpha \leq -2$ be an integer. Prove that for every pair $(\beta_0, \beta_1)$ of integers there exists a uniquely determined sequence $0\leq q_0, \ldots, q_k<\alpha ^ 2 - \alpha$ of integers, such that $q_k\neq 0$ if $(\beta_0, \beta 1)\neq (0,0)$ and
$$\beta_i=\sum_{j=0}^k q_j(\alpha - i)^j,\text{ for }i=0,1$$
Over the sides of an equilateral triangle with area $ 1$ are triangles with the opposite angle $ 60^{\circ}$ to each side drawn outside of the triangle. The new corners are $ P$, $ Q$ and $ R$. (and the new triangles $ APB$, $ BQC$ and $ ARC$)
1)What is the highest possible area of the triangle $ PQR$?
2)What is the highest possible area of the triangle whose vertexes are the midpoints of the inscribed circles of the triangles $ APB$, $ BQC$ and $ ARC$?
Let $n$ be a positive integer. Suppose that the following simultaneous equations
$$\begin{cases} \sin x_1 + \sin x_2+ ...+ \sin x_n = 0 \\
\sin x_1 + 2\sin x_2+ ...+ n \sin x_n = 100 \end{cases}$$
has a solution, where $x_1 x_2,.., x_n$ are the unknowns. Find the smallest possible positive integer $n$. Justify your answer.
Let $A$ be a finite set of positive numbers , $B=\{\frac{a+b}{c+d} |a,b,c,d \in A \}$.
Show that: $\left | B \right | \ge 2\left | A \right |^2-1 $,
where $|X| $ be the number of elements of the finite set $X$.
(High School Affiliated to Nanjing Normal University )
A natural number $A$ is given. One may add to it one of its divisors $d$ ($1 < d < A$). One may then repeat this operation with the new number $A + d$ and so on. Prove that starting from $A = 4$ one can get any composite number by these operations.
(M Vyalyi)
Find all positive integers $n$ such that $$\sum_{d|n, 1\leq d <n}d^2=5(n+1)$$
Let $ ABC$ be a triangle with sides $ BC\equal{}a, CA\equal{}b,AB\equal{}c$ and let $ D$ and $ E$ be the midpoints of $ AC$ and $ AB$, respectively. Prove that the medians $ BD$ and $ CE$ are perpendicular to each other if and only if $ b^2\plus{}c^2\equal{}5a^2$.
Let $f_1$, $f_2$, $f_3$, ..., be a sequence of numbers such that
\[
f_n = f_{n - 1} + f_{n - 2}
\]
for every integer $n \ge 3$. If $f_7 = 83$, what is the sum of the first 10 terms of the sequence?
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.
Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.
If $p,q\in\mathbb{R}[x]$ satisfy $p(p(x))=q(x)^2$, does it follow that $p(x)=r(x)^2$ for some $r\in\mathbb{R}[x]$?
Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number.
Let $O$ be the intersection point of the diagonals of a parallelogram $ABCD$ . Prove that if the line $BC$ is tangent to the circle passing through the points $A, B$, and $O$, then the line $CD$ is tangent to the circle passing through the points $B, C$ and $O$.
(A Zaslavskiy)
Find all nonnegative real numbers $ x$ for which $ \sqrt[3]{13\plus{}\sqrt{x}}\plus{}\sqrt[3]{13\minus{}\sqrt{x}}$ is an integer.
The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$
[asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy]
$\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi(3\sqrt{3} +2) \qquad\textbf{(D) } 10 \pi (\sqrt{3} - 1) \qquad\textbf{(E) } \pi(\sqrt{3} + 6)$
An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?
Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.
A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled
i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$,
ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry?
[i]Bulgaria[/i]
Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .
$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position.
(Sergey Dorichenko)
Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively.
Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.
Pam lists the four smallest positive prime numbers in increasing order. When she divides the positive integer $N$ by the first prime, the remainder is $1$. When she divides $N$ by the second prime, the remainder is $2$. When she divides $N$ by the third prime, the remainder is $3$. When she divides $N$ by the fourth prime, the remainder is $4$. Find the least possible value for $N$.
How many different isosceles triangles have integer side lengths and perimeter 23?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11$