Found problems: 85335
Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?
Find all integers $ k$ such that for infinitely many integers $ n \ge 3$ the polynomial
\[ P(x) =x^{n+ 1}+ kx^n - 870x^2 + 1945x + 1995\]
can be reduced into two polynomials with integer coefficients.
For a positive integer $n$, let $A_n$ be the set of quadruplets $(a,b,c,d)$ of integers, satisfying the following properties simultaneously:
[list]
[*] $0\le a\le c\le n,$
[*] $0\le b\le d\le n,$
[*] $c+d>n,$ and
[*] $bc=ad+1.$
[/list]
Moreover, define
$$\alpha_n=\sum_{(a,b,c,d)\in A_n}\frac{1}{ab+cd}.$$
Find all real numbers $t$ such that $\alpha_n>t$ for every positive integer $n$.
You have some cyan, magenta, and yellow beads on a non-reorientable circle, and you can perform only the following operations:
1. Move a cyan bead right (clockwise) past a yellow bead, and turn the yellow bead magenta.
2. Move a magenta bead left of a cyan bead, and insert a yellow bead left of where the magenta bead ends up.
3. Do either of the above, switching the roles of the words ``magenta'' and ``left'' with those of ``yellow'' and ``right'', respectively.
4. Pick any two disjoint consecutive pairs of beads, each either yellow-magenta or magenta-yellow, appearing somewhere in the circle, and swap the orders of each pair.
5. Remove four consecutive beads of one color.
Starting with the circle: ``yellow, yellow, magenta, magenta, cyan, cyan, cyan'', determine whether or not you can reach
a) ``yellow, magenta, yellow, magenta, cyan, cyan, cyan'',
b) ``cyan, yellow, cyan, magenta, cyan'',
c) ``magenta, magenta, cyan, cyan, cyan'',
d) ``yellow, cyan, cyan, cyan''.
[i]Proposed by Sammy Luo[/i]
Show that for each positive integer $n$, there is a positive integer $k$ such that the decimal representation of each of the numbers $k, 2k,\ldots, nk$ contains all of the digits $0, 1, 2,\ldots, 9$.
The circle inscribed in the triangle $A_1A_2A_3$ is tangent to its sides $A_1A_2, A_2A_3, A_3A_1$ at points $T_1, T_2, T_3$, respectively. Denote by $M_1, M_2, M_3$ the midpoints of the segments $A_2A_3, A_3A_1, A_1A_2$, respectively. Prove that the perpendiculars through the points $M_1, M_2, M_3$ to the lines $T_2T_3, T_3T_1, T_1T_2$ meet at one point.
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $ 75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?
$ \textbf{(A)}\ 2: 1 \qquad
\textbf{(B)}\ 3: 1 \qquad
\textbf{(C)}\ 4: 1 \qquad
\textbf{(D)}\ 16: 3 \qquad
\textbf{(E)}\ 6: 1$
An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?
Find the number of solutions to the given congruence$$x^{2}+y^{2}+z^{2} \equiv 2 a x y z \pmod p$$ where $p$ is an odd prime and $x,y,z \in \mathbb{Z}$.
[i]Proposed by math_and_me[/i]
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying
$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.
In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .
Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.
Let $f\colon \mathbb R ^2 \rightarrow \mathbb R$ be given by $f(x,y)=(x^2-y^2)e^{-x^2-y^2}$.
a) Prove that $f$ attains its minimum and its maximum.
b) Determine all points $(x,y)$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0$ and determine for which of them $f$ has global or local minimum or maximum.
In the country some mathematicians know each other and any division of them into two sets contain 2 friends from different sets.It is known that if you put any set of four or more mathematicians at a round table so that any two neighbours know each other , then at the table there are two friends not sitting next to each other.We denote by $c_i $ the number of sets of $i$ pairwise familiar mathematicians(by saying "familiar" it means know each other).Prove that
$c_1-c_2+c_3-c_4+...=1$
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
Find all real numbers $x$ which satisfied $|2x+1|+|x-1|=2-x$ .
The tennis federation has assigned numbers to $1024$ sportsmen, participating in the tournament, according to their skill. (The tennis federation uses the olympic system of tournaments. The looser in the pair leaves, the winner meets with the winner of another pair. Thus, in the second tour remains $512$ participants, in the third -- $256$, et.c. The winner is determined after the tenth tour.) It comes out, that in the play between the sportsmen whose numbers differ more than on $2$ always win that whose number is less. What is the greatest possible number of the winner?
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
Let $p>10$ be a prime. Prove that there is positive integers $m,n$ with $m+n<p$ such that $p$ divides $5^m7^n -1$
Consider the L-shaped tromino below with 3 attached unit squares. It is cut into exactly two pieces of equal area by a line segment whose endpoints lie on the perimeter of the tromino. What is the longest possible length of the line segment?
[asy]
size(1cm);
draw((0,0) -- (2,0) -- (2,1) -- (1,1)--(1,2)--(0,2)--(0,0));
[/asy]
[i]Proposed by James Lin.[/i]
Let $a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1$.
a) Show that $\sum\limits_{n=1}^{+\infty}a_nx^n$ converges for all $x \in (-4, 4)$ and that the function $f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n$ satisfies the differential equation $x(x - 4)f'(x) + (x + 2)f(x) = -x$.
b) Prove that $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$.
Determine all integers $n\geqslant 3$ such that there exist $n{}$ pairwise distinct real numbers $a_1,\ldots,a_n$ for which the sums $a_i+a_j$ over all $1\leqslant i<j\leqslant n$ form an arithmetic progression.
The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$. Let $k=a-1$. If the $k$-th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$, find the highest possible value of $n$.
Evaluate the expression
\[
\left|
\prod_{k=0}^{15} \left( 1+e^{2\pi i k^2/{31}} \right)
\right| \, .
\]
Let $BL, AD$ be the bisector and the altitude correspondingly of an acute triangle ABC. They intersect at point $T$. It turned out that the altitude $LK$ of $\triangle ALB$ is divided in half by the line $AD$. Prove that $KT \perp BL$.
[i]Proposed by Mariia Rozhkova[/i]