Found problems: 85335
A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
For each positive integer $n$, set $x_n=\binom{2n}{n}$.
a. Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$.
b. Find all positive integer $h>1$ such that there exists positive integers $N,T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$.
We represent the number line $R$ as the union of two non-empty sets $A, B$ different from $R$. Prove that one of the sets $A, B$ does not have the following property: the difference of any elements of the set belongs to the same set.
Is it possible to cut a square into five squares?
Let $S$ be a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can have?
The factors of $ x^4\plus{}64$ are:
$ \textbf{(A)}\ (x^2\plus{}8)^2 \qquad
\textbf{(B)}\ (x^2\plus{}8)(x^2\minus{}8) \qquad
\textbf{(C)}\ (x^2\plus{}2x\plus{}4)(x^2\minus{}8x\plus{}16) \\
\textbf{(D)}\ (x^2\minus{}4x\plus{}8)(x^2\minus{}4x\minus{}8) \qquad
\textbf{(E)}\ (x^2\minus{}4x\plus{}8)(x^2\plus{}4x\plus{}8)$
If $x, y, z$ are positive real numbers such that $x + y + z = 9xyz.$ Prove that:
$$\frac {x} {\sqrt {x^2+2yz+2}}+\frac {y} {\sqrt {y^2+2zx+2}}+\frac {z} {\sqrt {z^2+2xy+2}}\ge 1.$$
Consider when equality applies.
With three different digits, all greater than $0$, six different three-digit numbers are formed. If we add these six numbers together the result is $4.218$. The sum of the three largest numbers minus the sum of the three smallest numbers equals $792$. Find the three digits.
Let $n$ be the least possible value of $$\sqrt{x^2+y^2-2x+6y+19}+\sqrt{x^2+y^2+8x-4y+21}.$$ Find $n^2.$
Let $ABC$ be a triangle and $I$ its incenter. The point $D$ is on segment $BC$ and the circle $\omega$ is tangent to the circumcirle of triangle $ABC$ but is also tangent to $DC, DA$ at $E, F$, respectively. Prove that $E, F$ and $I$ are collinear.
We say that three numbers are balanced if either all three numbers are the same, or they are all different. A grid consisting of hexagons is presented in Figure 1. Each hexagon is filled with the number 1, 2, or 3, so that for any three hexagons that are mutually adjacent and oriented with two hexagons on the bottom and one hexagon on the top (as in Figure 3), the three numbers in the hexagons are balanced. Prove that when the grid is filled completely, the three numbers in the three shaded hexagons are balanced.
(An example of a partially filled-in grid is shown in Figure 2. There are other ways of filling in the grid.)
In January Jeff’s investment went up by three quarters. In February it went down by one quarter. In March it went up by one third. In April it went down by one fifth. In May it went up by one seventh. In June Jeff’s investment fell by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. If Jeff’s investment was worth the same amount at the end of June as it had been at the beginning of January, find $m + n$.
In a simple graph on $n$ vertices, every set of $k$ vertices has an odd number of common neighbours. Prove that $n+k$ must be odd.
Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities.
Remember that the number $1$ is not prime.
Are there integers $x,y$ and $z$ which satisfy the equation $$z^2=(x+1)(y^2-1)+n$$ when (a) $n=2006$ (b) $n=2007$?
Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$.
For positive integer $k$, define $x_k=3k+\sqrt{k^2-1}-2(\sqrt{k^2-k}+\sqrt{k^2+k})$. Then $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_{1681}}=\sqrt{m}-n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
The triangle $ABC$ is such that $\angle BAC=30^{\circ},\angle ABC=45^{\circ}$. Prove that if $X$ lies on the ray $AC$, $Y$ lies on the ray $BC$ and $OX=BY$, where $O$ is the circumcentre of triangle $ABC$, then $S_{XY}$ passes through a fixed point.
[i]Emil Kolev [/i]
Consider a circle tangent to sides $AB$ and $AC$ (these sides are not equal) of triangle $ABC$ and the circumscribed circle around it. Let $K$, $M$ and $P$ be the touchpoints of this circle with the sides of the triangle and with the circle circumscribed around it, respectively, and let $L$ be the midpoint of the arc $BC$ (not containing $A$). Prove that the lines $KM$, $PL$ and $BC$ intersect at one point.
Find the sum of all $5$-digit positive integers that they have only the digits $1, 2$, and $5$, none repeated more than three consecutive times.
There is convex $n-$gon. We color all its sides and also diagonals, that goes out from one vertex. So we have $2n-3$ colored segments. We write positive numbers on colored segments. In one move we can take quadrilateral $ABCD$ such, that $AC$ and all sides are colored, then remove $AC$ and color $BD$ with number $\frac{xz+yt}{w}$, where $x,y,z,t,w$ - numbers on $AB,BC,CD,DA,AC$. After some moves we found that all colored segments are same that was at beginning. Prove, that they have same number that was at beginning.
Let $p_1, p_2, . . . , p_{30}$ be a permutation of the numbers $1, 2, . . . , 30.$ For how many permutations does the equality $\sum^{30}_{k=1}|p_k - k| = 450 $ hold?
Let $k, n$ be odd positve integers greater than $1$. Prove that if there a exists natural number $a$ such that $k|2^a+1, \ n|2^a-1$, then there is no natural number $b$ satisfying $k|2^b-1, \ n|2^b+1$.
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
[i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]