Found problems: 85335
Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right
Proposed by: I.Kukharchuk
If $a @ b = \frac{a\times b}{a+b}$, for $a,b$ positive integers, then what is $5 @10$?
$\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$
Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon.
(V Proizvolov)
Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$. The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$. The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$. Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$. Prove that $A$, $E$, and $C$ are collinear if and only if $B$, $F$, and $D$ are collinear.
A number is called [i]nillless [/i] if it is integer and positive and contains no zeros. You can make a positive integer nillless by simply omitting the zeros. We denote this with square brackets, for example $[2050] = 25$ and $[13] = 13$. When we multiply, add, and subtract we indicate with square brackets when we omit the zeros. For example, $[[4 \cdot 5] + 7] = [[20] + 7] = [2 + 7] = [9] = 9$ and $[[5 + 5] + 9] = [[10] + 9] = [1 + 9] = [10] = 1$. The following is known about the two numbers $a$ and $b$:
$\bullet$ $a$ and $b$ are nillless,
$\bullet$ $1 < a < b < 100$,
$\bullet$ $[[a \cdot b] - 1] = 1$.
Which pairs $(a, b)$ satisfy these three requirements?
Let $k \in Z_{>0}$ be the smallest positive integer with the property that $k\frac{gcd(x,y)gcd(y,z)}{lcm (x,y^2,z)}$ is a positive integer for all values $1 \le x \le y \le z \le 121$. If k' is the number of divisors of $k$, find the number of divisors of $k'$.
Evaluate $1 + 2 + 4 + 7$
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$.
[i]Proposed by Mohammad Ahmadi[/i]
There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?
[b]a)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n$?
[b]b)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n^2$?
[i]Proposed by Morteza Saghafian[/i]
Perimeter of triangle $ABC$ is $4$. Point $X$ is marked at ray $AB$ and point $Y$ is marked at ray $AC$ such that $AX=AY=1$. Line segments $BC$ and $XY$ intersectat point $M$. Prove that perimeter of one of triangles $ABM$ or $ACM$ is $2$.
(V. Shmarov).
Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$.
Determine if there exist five consecutive positive integers such that their LCM is a perfect square.
In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$?
[img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]
A convex polygon $\mathcal{P}$ in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon $\mathcal{P}$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
In a $9\times 9$ table, all cells contain zeros. The following operations can be performed on the table:
1. Choose an arbitrary row, add one to all the numbers in that row, and shift all these numbers one cell to the right (and place the last number in the first position).
2. Choose an arbitrary column, subtract one from all its numbers, and shift all these numbers one cell down (and place the bottommost number in the top cell).
Is it possible to obtain a table in which all cells, except two, contain zeros, with 1 in the bottom-left cell and -1 in the top-right cell after several such operations?
[i]Proposed by N. Vlasova[/i]
Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality
\[
\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
\]
Let $n\ge 3$ be an integer. An [i]inner diagonal[/i] of a [i]simple $n$-gon[/i] is a diagonal that is contained in the $n$-gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$-gon $P$ and by $D(n)$ the least possible value of $D(Q)$, where $Q$ is a simple $n$-gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P)=D(n)$.
[i]Remark:[/i] A simple $n$-gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.
The altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.
Let $ f:[2,\infty )\rightarrow\mathbb{R} $ be a differentiable function satisfying $ f(2)=0 $ and
$$ \frac{df}{dx}=\frac{2}{x^2+f^4{x}} , $$
for any $ x\in [2,\infty ) . $ Show that there exists $ \lim_{x\to\infty } f(x) $ and is at most $ \ln 3. $
[i]Gabriel Daniilescu[/i]
Let $c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}$. \\Prove that at least one of $a,b$ is bigger than $\sqrt[11]{2015}$.
Given a prime $p$ and positive integer $k$, an integer $n$ with $0 \le n < p$ is called a $(p, k)$-Hofstadterian residue if there exists an infinite sequence of integers $n_0, n_1, n_2, \ldots$ such that $n_0 \equiv n$ and $n_{i + 1}^k \equiv n_i \pmod{p}$ for all integers $i \ge 0$. If $f(p, k)$ is the number of $(p, k)$-Hofstadterian residues, then compute $\displaystyle \sum_{k = 1}^{2016} f(2017, k)$.
[i]Proposed by Ashwin Sah[/i]
Consider a square painting of size $1 \times 1$. A rectangular sheet of paper of area $2$ is called its “envelope” if one can wrap the painting with it without cutting the paper. (For instance, a $2 \times 1$ rectangle and a square with side $\sqrt2$ are envelopes.)
a) Show that there exist other envelopes. (4)
b) Show that there exist infinitely many envelopes. (3)
How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips.
Let $ABC$ be a triangle with $\angle A = 80^o$ and $\angle C = 30^o$. Consider the point $M$ inside the triangle $ABC$ so that $\angle MAC= 60^o$ and $\angle MCA = 20^o$. If $N$ is the intersection of the lines $BM$ and $AC$ to show that a $MN$ is the bisector of the angle $\angle AMC$.