Found problems: 85335
Solve the system of equations: $\begin{cases}
x_1 + x_2 + x_3 = 6 \\
x_2 + x_3 + x_4 = 9 \\
x_3 + x_4 + x_5 = 3 \\
x_4 + x_5 + x_6 = -3 \\
x_5 + x_6 + x_7 = -9 \\
x_6 + x_7 + x_8 = -6 \\
x_7 + x_8 + x_1 = -2 \\
x_8 + x_1 + x_2 = 2 \end{cases}$
Alfonso teaches Francis how to draw a spiral in the plane: First draw half of a unit circle. Starting at one of the ends, draw half a circle with radius $1/2$. Repeat this process at the endpoint of each half circle, where each time the radius is half of the previous half-circle. Assuming you can’t stop Francis from drawing the entire spiral, compute the total length of the spiral.
In a convex $n$-gon all angles are equal from a certain point, located inside the $n$-gon, all its sides are seen under equal angles. Can we conclude that this $n$-gon is regular?
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]
Let $N$ be the number of $105$-digit positive integers that contain the digit 1 an odd number of times. Find the remainder when $N$ is divided by $1000$.
[i]Proposed by Harry Kim[/i]
Let $\ell$ be a given line, $A$ and $B$ given points of the plane. Choose a point $P$ on $\ell $ so that the longer of the segments $AP$, $BP$ is as short as possible. (If $AP = BP,$ either segment may be taken as the longer one.)
Let $ABCD$ be a cyclic quadrilateral with $O$ is circumcenter and $AC$ meets $BD$ at $I$ Suppose that rays $DA,CD$ meet at $E$ and rays $BA,CD$ meet at $F$. The Gauss line of $ABCD$ meets $AB,BC,CD,DA$ at points $M,N,P,Q$ respectively. Prove that the circle of diameter $OI$ is tangent to two circles $(ENQ), (FMP)$
It is known that in the triangle $ABC$ the smallest side is $BC$. Let $X, Y, K$ and $L$ - points on the sides $AB, AC$ and on the rays $CB, BC$, respectively, are such that $BX = BK = BC =CY =CL$. The line $KX$ intersects the line $LY$ at the point $M$. Prove that the intersection point of the medians $\vartriangle KLM$ coincides with the center of the inscribed circle $\vartriangle ABC$.
We define the sets of lattice points $S_0,S_1,\ldots$ as $S_0=\{(0,0)\}$ and $S_k$ consisting of all lattice points that are exactly one unit away from exactly one point in $S_{k-1}$. Determine the number of points in $S_{2017}$.
[i]Proposed by Michael Ren
Find all positive integers $a,n\ge1$ such that for all primes $p$ dividing $a^n-1$, there exists a positive integer $m<n$ such that $p\mid a^m-1$.
Let $\omega_1$ and $\omega_2$ be two circles with no common points, such that any of them is not inside the other one. Let $M, N$ lie on $\omega_1, \omega_2$, such that the tangents at $M$ to $\omega_1$ and $N$ to $\omega_2$ meet at $P$, such that $PM=PN$. The circles $\omega_1$, $\omega_2$ meet $MN$ at $A, B$. The lines $PA, PB$ meet $\omega_1, \omega_2$ at $C, D$. Show that $\angle BCN=\angle ADM$.
Let $m$ and $n$ be arbitrary positive integers, and
$a, b, c$ be different natural numbers of the form $2^m.5^n$. Determine
the number of all equations of the form $ax^2-2bx+c=0$ if it is known that each equation
has only one real solution.
Let $a_n = n(2n+1)$. Evaluate
\[
\biggl | \sum_{1 \le j < k \le 36} \sin\bigl( \frac{\pi}{6}(a_k-a_j) \bigr) \biggr |.
\]
Alice and Bob play a number game. Starting with a positive integer $n$ they take turns changing the number with Alice going first. Each player may change the current number $k$ to either $k-1$ or $\lceil k/2\rceil$. The person who changes $1$ to $0$ wins. Determine all $n$ such that Alice has a winning strategy.
Assume that $ a$, $ b$, $ c$ and $ d$ are the sides of a quadrilateral inscribed in a given circle. Prove that the product $ (ab \plus{} cd)(ac \plus{} bd)(ad \plus{} bc)$ acquires its maximum when the quadrilateral is a square.
On an in finite chessboard are placed $2009 \ n \times n$ cardboard pieces such that each of them covers exactly $n^2$ cells of the chessboard. Prove that the number of cells of the chessboard which are covered by odd numbers of cardboard pieces is at least $n^2.$
[i](9 points)[/i]
Does there exist a set $S$ of positive integers satisfying the following conditions?
$\text{(i)}$ $S$ contains $2020$ distinct elements;
$\text{(ii)}$ the number of distinct primes in the set $\{\gcd(a, b) : a, b \in S, a \neq b\}$ is exactly $2019$; and
$\text{(iii)}$ for any subset $A$ of $S$ containing at least two elements, $\sum\limits_{a,b\in A; a<b}
ab$ is not a prime power.
Suppose that $f(x)=x^2-10x+21$. Find all distinct real roots of $f(f(x)+7)$.
Calculate the integral
\[\int_{-\infty}^{+\infty}e^{ikx}\frac{1-e^x}{1+e^x}dx\]
A circle of radius $ r$ is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is 1/2. What is $ r$?
$ \textbf{(A) } 2\sqrt {2} \plus{} 2\sqrt {3} \qquad \textbf{(B) } 3\sqrt {3} \plus{} \sqrt {2} \qquad \textbf{(C) } 2\sqrt {6} \plus{} \sqrt {3} \qquad \textbf{(D) } 3\sqrt {2} \plus{} \sqrt {6}\\
\textbf{(E) } 6\sqrt {2} \minus{} \sqrt {3}$
An equilateral triangle whose side is $ 2$ is divided into a triangle and a trapezoid by a line drawn parallel to one of its sides. If the area of the trapezoid equals one-half of the area of the original triangle, the length of the median of the trapezoid is:
$ \textbf{(A)}\ \frac{\sqrt{6}}{2} \qquad
\textbf{(B)}\ \sqrt{2} \qquad
\textbf{(C)}\ 2\plus{}\sqrt{2} \qquad
\textbf{(D)}\ \frac{2\plus{}\sqrt{2}}{2} \qquad
\textbf{(E)}\ \frac{2\sqrt{3}\minus{}\sqrt{6}}{2}$
Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the centre $O$ lies between these chords. Suppose $AB = 6, CD = 8$. Suppose further that the area of the part of the circle lying between the chords $AB$ and $CD$ is $(m\pi + n) / k$, where $m, n, k$ are positive integers with gcd$(m, n, k) = 1$. What is the value of $m + n + k$ ?
[b]p22.[/b] Find the unique ordered pair $(m, n)$ of positive integers such that $x = \sqrt[3]{m} -\sqrt[3]{n}$ satisfies $x^6 + 4x^3 - 36x^2 + 4 = 0$.
[b]p23.[/b] Jenny plays with a die by placing it flat on the ground and rolling it along any edge for each step. Initially the face with $1$ pip is face up. How many ways are there to roll the dice for $6$ steps and end with the $1$ face up again?
[b]p24.[/b] There exists a unique positive five-digit integer with all odd digits that is divisible by $5^5$. Find this integer.
PS. You should use hide for answers.
Let $w\in \mathbb{C}\setminus \mathbb{R}$, $|w|\neq 1$. Prove that $f\colon \mathbb{C} \to \mathbb{C}$, given by $f(z)= z+w\overline{z}$, is a bijection, and find its inverse.
There is a rhombus with acute angle $b$ and side $a$. Two parallel lines, the distance between which is equal to the height of the rhombus, intersect all four sides of the rhombus. What can be the sum of the perimeters of two triangles cut off from a rhombus by straight lines? (These two triangles lie outside the strip between parallel lines.)