Found problems: 85335
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
Let $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \le n$, define $S_m=\displaystyle\sum_{\substack{i-j=m \\0 \le i,j \le n}} \dfrac{1}{2^{i+j}}$. Then \[\lim_{n\to \infty} \left( S_{-n}^2 + S_{-n+1}^2 +... + S_n^2 \right)\] can be expressed in the form $\dfrac{p}{q}$ for relatively prime positive integers $p,q$. Compute $100p+q$.
[i]Proposed by Vincent Huang[/i]
Let $\mathfrak F$ be the family of all $k$-element subsets of the set $\{1, 2, \ldots, 2k + 1\}$. Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$, the sets $A$ and $f(A)$ are disjoint.
Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.
$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?
Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying:
i) $f(1)=f(2)=1$;
ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$.
For each integer $m\ge 2$, find the value of $f(2^m)$.
There is a circle $c$ centered about the origin of radius $ 1$. There are circles $c_1$,$ . . .$ ,$c_6$, each of radius $r_1$, such that each circle is completely inside c and is tangent to it, and $c_2$ is tangent to $c_1$, $c_3$ is tangent to $c_2$, . . ., and $c_1$ is tangent to $c_6$. There is a circle $d$ which is tangent to $c$, $c_1$, and $c_2$, but does not intersect any of these circles. What is the radius of circle $d$? Express your answer in the form $\frac{a+b\sqrt{c}}{d}$ , where $a,b,c,d$ are integers, $d$ is positive and as small as possible, and $c$ is squarefree.
Let $a$ and $b$ be integers (not necessarily positive). Prove that $a^3+5b^3 \neq 2016$.
Prove that there exists a positive integer $n$, such that the remainder of $3^n$ when divided by $2^n$ is greater than $10^{2021} $.
For a positive integer $n$ denote $d(n)$ its greatest odd divisor. Find the value of the sum
$$d(1008)+d(1009)+...+d(2015)$$
Find all triples $(a, b, c)$ of positive integers to the equation \[a! b! = a!+b!+c!.\]
Find the number of sequences of $0, 1$ with length $n$ satisfying both of the following properties:
[list]
[*] There exists a simple polygon such that its $i$-th angle is less than $180$ degrees if and only if the $i$-th element of the sequence is $1$.
[*] There exists a convex polygon such that its $i$-th angle is less than $90$ degrees if and only if the $i$-th element of the sequence is $1$.
[/list]
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)\equal{}x^3\minus{}2007x\plus{}2002.$
Determine the value of: $ \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}$.
$n$ is a natural number, $f_n(x)=\frac{x^{n+1}-x^{-n-1}}{x-x^{-1}}(x\neq0,\pm1)$, let $y=x+\frac{1}{x}$.
[b](a)[/b] Prove that $f_{n+1}(x)=yf_n(x)-f_{n-1}(x)$
[b](b)[/b] Prove with mathematical induction:
$f_n(x)=\begin{cases}
y^n-\text{C}_{n-1}^{1}y^{n-2}+\cdots+(-1)^i\text{C}_{n-i}^{i}y^{n-2i}+\cdots+(-1)^{\frac{n}{2}}(i=1,2,\cdots,\frac{n}{2},n\text{ is even})\\
y^n-\text{C}_{n-1}^{1}y^{n-2}+\cdots+(-1)^i\text{C}_{n-i}^{i}y^{n-2i}+\cdots+(-1)^{\frac{n-1}{2}}\text{C}_{\frac{n+1}{2}}^{\frac{n-1}{2}}y(i=1,2,\cdots,\frac{n-1}{2},n\text{ is odd})
\end{cases}$.
In convex quadrilateral $ABCD$, let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$, respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$, and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$. Prove that $A$, $B$, $R$, and $S$ are concyclic.
[i]Tiger Zhang[/i]
On a piece of paper we have $2019$ statements numbered from $1$ to $2019$. The $n^{th}$ statement is the following: "On this piece of paper there are at most $n$ true statements". How many of the statements are true?
On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?
$\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$
Find the smallest and largest integers with decimal representation of the form $ababa$ ($a \neq 0$) that are divisible by $11$.
Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?
In a acute-angled triangle $ABC$, a point $D$ lies on the segment $BC$. Let $O_1,O_2$ denote the circumcentres of triangles $ABD$ and $ACD$ respectively. Prove that the line joining the circumcentre of triangle $ABC$ and the orthocentre of triangle $O_1O_2D$ is parallel to $BC$.
We define a sequence of numbers $a_n$ such that $a_0=1$ and for all $n\ge0$:
\[2a_{n+1} ^3 + 2a_n ^3 = 3 a_{n +1} ^2 a_n + 3a_{n+1}a_n^2\]
Find the sum of all $a_{2023}$'s possible values.
Decide whether it is possible to color the $1984$ natural numbers $1, 2, 3, \cdots, 1984$ using $15$ colors so that no geometric sequence of length $3$ of the same color exists.
Let $n\geq 3$ be an integer and $S_{n}$ the permutation group. $G$ is a subgroup of $S_{n}$, generated by $n-2$ transpositions. For all $k\in\{1,2,\ldots,n\}$, denote by $S(k)$ the set $\{\sigma(k) \ : \ \sigma\in G\}$.
Show that for any $k$, $|S(k)|\leq n-1$.