Found problems: 85335
There are $24$ robots on the plane. Each robot has a $70^{\circ}$ field of view. What is the maximum number of observing relations?
(Observing is a one-sided relation)
Let $ G$ be a simple graph with $ 2 \cdot n$ vertices and $ n^{2}+1$ edges. Show that this graph $ G$ contains a $ K_{4}-\text{one edge}$, that is, two triangles with a common edge.
A circle $k$ touches a larger circle $K$ from inside in a point $P$. Let $Q$ be point on $k$ different from $P$. The line tangent to $k$ at $Q$ intersects $K$ in $A$ and $B$.
Show that the line $PQ$ bisects $\angle APB$.
In triangle $ABC$, $\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$. Find $CE$ given that $AC=35$, $BC=7$, and $BE=5$.
If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$,
$$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$
find the smallest $n$ such that $a_n < \frac{1}{2018}$.
Let $a,b$ be coprime positive integers with $a$ even and $a>b$. Show that there exist infinitely many pairs $(m,n)$ of coprime positive integers such that $m\mid a^{n-1}-b^{n-1}$ and $n\mid a^{m-1}-b^{m-1}$.
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum.
Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)
Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.
Find all prime numbers $p$ and nonnegative integers $x\neq y$ such that $x^4- y^4=p(x^3-y^3)$.
[i]Proposed by Bulgaria[/i]
In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are rational numbers, we call it rational point. If $a$ is an irrational number, then in all lines that passes $(a,0)$,
$\text{(A)}$There are infinitely many lines, on which there are at least two rational points.
$\text{(B)}$There are exactly $n(n\geq2)$ lines, on which there are at least two rational points.
$\text{(C)}$There are exactly 1 line, on which there are at least two rational points.
$\text{(D)}$Every line passes at least one rational point.
In a "Hanger Party", the guests are initially dressed. In certain moments, the host chooses a guest, and the chosen guest and all his friends will wear its respective clothes if they are naked, and undress it if they are dressed.
It is possible that, in some moment, the guests are naked, independent of their mutual friendships? (Suppose friendship is reciprocal.)
Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$.
($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)
Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n-1)(b^n-1)$ is not a perfect square.
[i]Proposed by Mongolia[/i]
A frog starts a journey at $(6,9).$ A skip is the act of traveling a positive integer number of units straight south or a positive integer number of units straight west. A jump is the act of traveling one unit straight west. A hop consists of any skip followed by a jump. How many different sequences of hops can the frog take so that the frog's final destination is $(0,0)$?
[i]Proposed by Jack Ma[/i]
Let $n$ be a natural number. Prove that a necessary and sufficient condition for the equation $z^{n+1}-z^n-1=0$ to have a complex root whose modulus is equal to $1$ is that $n+2$ is divisible by $6$.
There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 11$, has a remainder of $1$. What is the difference between the two smallest such integers?
$\textbf{(A) }2310\qquad\textbf{(B) }2311\qquad\textbf{(C) }27,720\qquad\textbf{(D) }27,721\qquad \textbf{(E) }\text{none of these}$
The sum of two integers is $8$. The sum of the squares of those two integers is $34$. What is the product of the two
integers?
Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb R $ to $\mathbb R $ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $f(x)-g(x)$ is an integer.
Circle $\odot O$ with diameter $\overline{AB}$ has chord $\overline{CD}$ drawn such that $\overline{AB}$ is perpendicular to $\overline{CD}$ at $P$. Another circle $\odot A$ is drawn, sharing chord $\overline{CD}$. A point $Q$ on minor arc $\overline{CD}$ of $\odot A$ is chosen so that $\text{m} \angle AQP + \text{m} \angle QPB = 60^\circ$. Line $l$ is tangent to $\odot A$ through $Q$ and a point $X$ on $l$ is chosen such that $PX=BX$. If $PQ = 13$ and $BQ = 35$, find $QX$.
[i]Proposed by Aaron Lin[/i]
Let $ O $ be the circumcenter of a triangle $ ABC $ with $ \angle BAC=60^{\circ } $ whose incenter is denoted by $ I. $ Let $ B_1,C_1 $ be the intersection of $ BI,CI $ with the circumcircle of $ ABC, $ respectively. Denote by $ O_1,O_2 $ the circumcenters of $ BIC_1,CIB_1, $ respectively. Show that $ O_1,I,O,O_2 $ are collinear.
[i]Cătălin Țigăeru[/i]
Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.
Gary has $2$ children. We know one is a boy born on a Friday. Assume birthing boys and girls are equally likely, being born on any day of the week is equally likely, and that these properties are independent of each other, as well as independent from child to child. The probability that both of Gary's children are boys is $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime integers. Find $a + b.$
In an acute non-isosceles triangle $ABC$, the altitude $AA'$ is drawn and point $H$ is the intersection point of the altitudes and and $O$ is the center of the circumscribed circle. Prove that the point symmetric to the circumcenter of triangle $HOA'$ wrt straight line $HO$, lies on a midline of triangle $ABC$.
$ 2^n $ coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for interchange. What is the greatest possible number of consecutive interchanges? ($ n $ is natural number)
A box contains answer $4032$ scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among
which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left.