Found problems: 85335
Show that for any positive integers $k$ and $1 \leq a \leq 9$, there exists $n$ such that satisfies the below statement.
When $2^n=a_0+10a_1+10^2a_2+ \cdots +10^ia_i+ \cdots $ $(0 \leq a_i \leq 9$ and $a_i$ is integer), $a_k$ is equal to $a$.
Compute the sum of the distinct prime factors of $10101$.
[i]Proposed by Lewis Chen[/i]
In convex pentagon $ABCDE, AC$ is parallel to $DE, AB$ is perpendicular to $AE$, and $BC$ is perpendicular to $CD$. If $H$ is the orthocentre of triangle $ABC$ and $M$ is the midpoint of segment $DE$, prove that $AD, CE$ and $HM$ are concurrent.
Let $n$ be a positive integer. Solve the equation in $\mathbb{R}$ $$\sqrt[2n+1]{x}+\sqrt[2n+1]{x+1}+\sqrt[2n+1]{x+2}+\dots+\sqrt[2n+1]{x+n}=0.$$
Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$.
Prove that if a triangle has integer side lengths and the area (in square units) equals the perimeter (in units), then the perimeter is not a prime number.
Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$, respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced.
Prove that $QO$ is perpendicular to $BC$.
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?
The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values.
[i]Proposed by A. Golovanov[/i]
Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$.
[i]Proposed by I. Dorofeev[/i]
Ria has $4$ green marbles and 8 red marbles. She arranges them in a circle randomly, if the probability that no two green marbles are adjacent is $\frac{p}{q}$ where the positive integers $p,q$ have no common factors other than $1$, what is $p+q$?
Given the true statement: The picnic on Sunday will not be held only if the weather is not fair. We can then conclude that:
$ \textbf{(A)}\ \text{If the picnic is held, Sunday's weather is undoubtedly fair.}$
$ \textbf{(B)}\ \text{If the picnic is not held, Sunday's weather is possibly unfair.}$
$ \textbf{(C)}\ \text{If it is not fair Sunday, the picnic will not be held.}$
$ \textbf{(D)}\ \text{If it is fair Sunday, the picnic may be held.}$
$ \textbf{(E)}\ \text{If it is fair Sunday, the picnic must be held.}$
Given a regular hexagon, a circle is drawn circumscribing it and another circle is drawn inscribing it. The ratio of the area of the larger circle to the area of the smaller circle can be written in the form $\frac{m}{n}$ , where m and n are relatively prime positive integers. Compute $m + n$.
A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.
Find all functions $f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}$ that satisfy $f(mf(n)) = n+f(2015m)$ for all $m,n \in \mathbb{Z}_{+}$.
Given a 4×4 grid of points, the points at two opposite corners are denoted $A$ and $D$. We need to choose two other points $ B$ and $C$ such that the six pairwise distances of these four points are all distinct.
(a) How many such quadruples of points are there?
(b) How many such quadruples of points are non-congruent?
(c) If each point is assigned a pair of coordinates $(x_i,y_i)$, prove that the sum of the expressions $|x_i-x_j |+|y_i-y_j|$ over all six pairs of points in a quadruple is constant.
Find all pairs ($x,y$) of integers such that $x^2-2xy+126y^2=2009$.
For what positive integers $n$ are there positive rational, but not integer, numbers $a$ and $b$ such that both numbers $a + b$ and $a^n + b^n$ are integers?
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.
Prove that if the equation $$x^4 + ax + b = 0$$ has two equal roots, then $$\left( \frac{a}{4} \right)^4 =\left( \frac{b}{3} \right)^3.$$
Let $P$ and $Q$ be points selected uniformly and independently at random inside a regular hexagon $ABCDEF.$ Compute the probability that segment $\overline{PQ}$ is entirely contained in at least one of the quadrilaterals $ABCD,$ $BCDE,$ $CDEF,$ $DEFA,$ $EFAB,$ or $FABC.$
Find, with proof, all positive integers $n$ with the following property: There are only finitely many positive multiples of $n$ which have exactly $n$ positive divisors
Show that if a triangle has two excircles of the same size, then the triangle is isosceles.
(Note: The excircle $ABC$ to the side $ a$ touches the extensions of the sides $AB$ and $AC$ and the side $BC$.)
Real numbers $x,y$ satisfy that $\begin{cases}
(x-1)^3+1997(x-1)=-1\\
(y-1)^3+1997(y-1)=1
\end{cases}$, then $x+y=$________.
\[\int_1^e \frac{\log(x^{2024})}{x} \mathrm dx\]
[i]Proposed by Connor Gordon[/i]