Found problems: 85335
Find the smallest positive integer $k$ such that
\[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\]
for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.)
[i]Alex Zhu.[/i]
[hide="Clarifications"][list=1][*]${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$. In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$.[/list][/hide]
All internal angles of a convex octagon $ABCDEFGH$ are equal to each other and the edges are alternatively equal:
$$AB = CD = EF = GH,BC = DE = FG = HA$$
(we call such an octagon semiregular). The diagonals $AD$, $BE$, $CF$, $DG$, $EH$, $FA$, $GB$ and $HC$ divide the inside of the octagon into certain parts. Consider the part containing the centre of the octagon. If that part is an octagon, then this central octagon is semiregular (this is obvious). In this case we construct similar diagonals in the central octagon and so on. If, after several steps, the central figure is not an octagon, then the process stops. Prove that if the process never stops, then the initial octagon was regular.
(A. Tolpygo, Kiev)
Points $ X $ and $ Y $ are the midpoints of the sides $ AB $ and $ AC $ of the triangle $ ABC $, $ I $ is the center of its inscribed circle, $ K $ is the point of tangency of the inscribed circles with side $ BC $. The external angle bisector at the vertex $ B $ intersects the line $ XY $ at the point $ P $, and the external angle bisector at the vertex of $ C $ intersects $ XY $ at $ Q $. Prove that the area of the quadrilateral $ PKQI $ is equal to half the area of the triangle $ ABC $.
Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.
Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.
Find all functions $ f :\mathbb{R}\to\mathbb{R} $, satisfying the condition
$f (xf (y) + f (x)) = 2f (x) + xy$
for any real $x$ and $y$.
Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.
The numbers $1, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2$ are written on the board. Is it possible to paint half of them red and remaining ones blue, so that the sum of red numbers is divisible by the sum of blue ones?
For the various values of the parameter $a \in R$, solve the equation $ ||x - 4| - 2x + 8| = ax + 4$
For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that:
a.) $N_a$ is odd;
b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.
For each positive integer $n$, let $\rho(n)$ be the number of positive divisors of $n$ with exactly the same set of prime divisors as $n$. Show that, for any positive integer $m$, there exists a positive integer $n$ such that $\rho(202^n+1)\geq m.$
The professor tells Peter the product of two positive integers and Sam their sum. At first, nobody of them knows the number of the other.
One of them says: [i]You can't possibly guess my number[/i].
Then the other says: [i]You are wrong, the number is 136[/i].
Which number did the professor tell them respectively? Give reasons for your claim.
In terms of $p$ and $k$, compute the number of solutions in positive integers to the equation
$ab+bc+ca=p^{2k}$ satisfying $a\leq b\leq c$ where $p$ is a fixed prime and $k$ is a fixed positive integer.
[i]Proposed by Alexander Wang[/i]
All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is [i]Extremadura[/i] if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of [i]Extremadura[/i] faces of this polyhedron is always even?
Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2$, $A_2A_3$,$ ...$, $A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$. Prove that it is possible
a) for $k=3$,
b) for every $k$ less than $(n-1)$.
Determine the number of digits $1$ in the integer part of $\frac{10^{1992}}{10^{83}+7}$.
Zan starts with a rational number $\tfrac{a}{b}$ written on the board in lowest terms. Then, every second, Zan adds $1$ to both the numerator and denominator of the latest fraction and writes the result in lowest terms. Zan stops as soon as he writes a fraction of the form $\tfrac{n}{n+1}$, for some positive integer $n$. If $\tfrac{a}{b}$ started in that form, Zan does nothing.
As an example, if Zan starts with $\tfrac{13}{19}$, then after one second he writes $\tfrac{14}{20} = \tfrac{7}{10}$, then after two seconds $\tfrac{8}{11}$, then $\tfrac{9}{12} = \tfrac{3}{4}$, at which point he stops.
(a) Prove that Zan will stop in less than $b-a$ seconds.
(b) Show that if $\tfrac{n}{n+1}$ is the final number, then \[\frac{n-1}{n} < \frac{a}{b} \le \frac{n}{n+1}.\]
[i](Proposed by Michael Tang.)[/i]
In a triangle $ABC$, two lines are drawn that together trisect the angle at $A$. These intersect the side $BC$ at points $P$ and $Q$ so that $P$ is closer to $B$ and $Q$ is closer to $C$. Determine the smallest constant $k$ such that $| P Q | \le k (| BP | + | QC |)$, for all such triangles. Determine if there are triangles for which equality applies.
A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.
Squares of an $8 \times 8$ chessboard are each allocated a number between $1$ and $32$ , with each number being used twice. Prove that it is possible to choose $32$ such squares, each allocated a different number, so that there is at least one such square on each row or column .
(A . Andjans, Riga
Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is:
a) 8 ;
6) 99 ?
Alexey Glebov
The angle and the point $K$ inside it are given on the plane. Prove that there is a point $M$ with the following property:
if an arbitrary line passing through intersects the sides of the angle at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.
Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that\[ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}\]
Let $ABCDEF$ be a convex hexagon that does not have two parallel sides, such that $\angle AF B = \angle F DE, \angle DF E = \angle BDC$ and $\angle BFC = \angle ADF.$ Prove that the lines $ AB, FC$ and $DE$ are concurrent if and only if the lines $ AF, BE$ and $CD$ are concurrent.
Let $\alpha$ be the positive root of the quadratic equation $x^2 = 1990x + 1$. For any $m, n \in \mathbb N$, define the operation $m*n = mn + [\alpha m][ \alpha n]$, where $[x]$ is the largest integer no larger than $x$. Prove that $(p*q)*r = p*(q*r)$ holds for all $p, q, r \in \mathbb N.$