Found problems: 1679
[color=darkred]Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .[/color]
A circle of radius $ 1$ is surrounded by $ 4$ circles of radius $ r$ as shown. What is $ r$?
[asy]defaultpen(linewidth(.9pt));
real r = 1 + sqrt(2);
pair A = dir(45)*(r + 1);
pair B = dir(135)*(r + 1);
pair C = dir(-135)*(r + 1);
pair D = dir(-45)*(r + 1);
draw(Circle(origin,1));
draw(Circle(A,r));draw(Circle(B,r));draw(Circle(C,r));draw(Circle(D,r));
draw(A--(dir(45)*r + A));
draw(B--(dir(45)*r + B));
draw(C--(dir(45)*r + C));
draw(D--(dir(45)*r + D));
draw(origin--(dir(25)));
label("$r$",midpoint(A--(dir(45)*r + A)), SE);
label("$r$",midpoint(B--(dir(45)*r + B)), SE);
label("$r$",midpoint(C--(dir(45)*r + C)), SE);
label("$r$",midpoint(D--(dir(45)*r + D)), SE);
label("$1$",origin,W);[/asy]$ \textbf{(A)}\ \sqrt {2}\qquad \textbf{(B)}\ 1 \plus{} \sqrt {2}\qquad \textbf{(C)}\ \sqrt {6}\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 2 \plus{} \sqrt {2}$
In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.
In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$.
[b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$.
[b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.
Triangle $ABC$ has $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$, and meets $BC$ at $M$. What is the nearest integer to $100 \frac{AM}{CM}$?
Let $ABC$ be an equilateral triangle of area $1998$ cm$^2$. Points $K, L, M$ divide the segments $[AB], [BC] ,[CA]$, respectively, in the ratio $3:4$ . Line $AL$ intersects the lines $CK$ and $BM$ respectively at the points $P$ and $Q$, and the line $BM$ intersects the line $CK$ at point $R$. Find the area of the triangle $PQR$.
Consider the maximum value of circular cone inscribed to a sphere, the ratio of it to the volume of the sphere is________.
In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.
The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$
by Mahdi Etesami Fard
A triangle ABC with $\angle A = 60^o$ is given. Points $M$ and $N$ on $AB$ and $AC$ respectively are such that the circumcenter of $ABC$ bisects segment $MN$. Find the ratio $AN:MB$.
by E.Bakaev
Excircle of triangle $ABC$ corresponding vertex $A$, is tangent to $BC$ at $P$. $AP$ intersects circumcircle of $ABC$ at $D$. Prove \[r(PCD)=r(PBD)\] whcih $r(PCD)$ and $r(PBD)$ are inradii of triangles $PCD$ and $PBD$.
From the vertex $C$ in triangle $ABC$, draw a straight line that bisects the median from $A$. In what ratio does this line divide the segment $AB$?
[img]https://1.bp.blogspot.com/-SxWIQ12DIvs/XzcJv5xoV0I/AAAAAAAAMY4/Ezfe8bd7W-Mfp2Qi4qE_gppbh9Fzvb4XwCLcBGAsYHQ/s0/1995%2BMohr%2Bp3.png[/img]
In triangle $ABC,$ point $D$ is on $\overline{BC}$ with $CD=2$ and $DB=5,$ point $E$ is on $\overline{AC}$ with $CE=1$ and $EA=3,$ $AB=8,$ and $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}.$ It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
Let $ABC$ be a triangle, let $B_{1}$ be the midpoint of the side $AB$ and $C_{1}$ the midpoint of the side $AC$. Let $P$ be the point of intersection, other than $A$, of the circumscribed circles around the triangles $ABC_{1}$ and $AB_{1}C$. Let $P_{1}$ be the point of intersection, other than $A$, of the line $AP$ with the circumscribed circle around the triangle $AB_{1}C_{1}$. Prove that $2AP=3AP_{1}$.
Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation
\[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\]
For all $x,y\in\mathbb R.$
Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.
$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$, where $1/2 < k < 1$. Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$, and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$. Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$.
Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$.
[img]https://3.bp.blogspot.com/-9EkSdxpGnPU/W6dWD82CxwI/AAAAAAAAJHw/iTkEOejlm9U6Dbu427vUJwKMfEOOVn0WwCK4BGAYYCw/s400/Yasinsky%2B2017%2BVIII-IX%2Bp1.png[/img]
Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?
On the sides $AB$ and $AC$ of a triangle $ABC$ select points $D$ and $E$ respectively, such that $AB = 6$, $AC = 9$, $AD = 4$ and $AE = 6$. It is known that the circumscribed circle of $\vartriangle ADE$ interects the side $BC$ at points $F, G$ , where $BF < BG$. Knowing that the point of intersection of lines $DF$ and $EG$ lies on the circumscribed circle of $\vartriangle ABC$ , find the ratio $BC:FG$.
Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, where $p$ and $q$ are relatively prime integers, find $p+q$.
Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10.
[i]L. Kuptsov[/i]
Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that
\[\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE} \]
The ratio of the interior angles of two regular polygons with sides of unit length is $ 3: 2$. How many such pairs are there?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ \text{infinitely many}$