Found problems: 1581
Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$.
(1) Find the maximum value of $ y$.
(2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis.
(3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.
Let $ABC$ be an acute triangle with incenter $I$. On its circumcircle, let $M_A$, $M_B$ and $M_C$ be the midpoints of minor arcs $BC, CA$ and $AB$, respectively. Prove that the reflection $M_A$ over the line $IM_B$ lies on the circumcircle of the triangle $IM_BM_C$.
Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.
Let squares be constructed on the sides $BC,CA,AB$ of a triangle $ABC$, all to the outside of the triangle, and let $A_1,B_1, C_1$ be their centers. Starting from the triangle $A_1B_1C_1$ one analogously obtains a triangle $A_2B_2C_2$. If $S, S_1, S_2$ denote the areas of triangles$ ABC,A_1B_1C_1,A_2B_2C_2$, respectively, prove that $S = 8S_1 - 4S_2.$
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.
[i]Proposed by David Stoner[/i]
Let $ABC$ be a triangle inscribed in a circle of radius $1$. If the triangle's sides are integer numbers, then find that triangle's sides.
Prove that the transformation product of the symmetry of center $(0, 0)$ with the symmetry of the axis, with the line of equation $x = y + 1$, can be expressed as a product of an axis symmetry the line $e$ by a translation of vector $\overrightarrow{v}$, with $e$ parallel to $\overrightarrow{v}$, .
Determine a line $e$ and a vector $\overrightarrow{v}$, that meet the indicated conditions. have to be unique $e$ and $\overrightarrow{v}$,?
It is known that the graph of the function $y = f(x)$ after a rotation of $45^o$ around a certain point turns into the graph of the function $y = x^3 + ax^2 + 19x + 97$. At what $a$ is this possible?
A square hole of depth $h$ whose base is of length $a$ is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length $L >\sqrt{2a^2+h^2}$, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide along it. Find the shape and area of the territory accessible to the dog (whose size is neglected).
Given a point $P$ on a triangular piece of paper $ABC,$ consider the creases that are formed in the paper when $A, B,$ and $C$ are folded onto $P.$ Let us call $P$ a fold point of $\triangle ABC$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,$ and $\angle B=90^\circ.$ Then the area of the set of all fold points of $\triangle ABC$ can be written in the form $q\pi-r\sqrt{s},$ where $q, r,$ and $s$ are positive integers and $s$ is not divisible by the square of any prime. What is $q+r+s$?
Let $I$ be the incenter of the scalene $\Delta ABC$, such, $AB<AC$, and let $I'$ be the reflection of point $I$ in line $BC$. The angle bisector $AI$ meets $BC$ at $D$ and circumcircle of $\Delta ABC$ at $E$. The line $EI'$ meets the circumcircle at $F$. Prove, that,
$\text{(i) } \frac{AI}{IE}=\frac{ID}{DE}$
$\text{(ii) } IA=IF$
In triangle $ABC,$ $AB=13,$ $BC=14,$ $AC=15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area if the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$
Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI = AR$.
In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$. Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$. Let us consider an arbitrary point $M$ (which is different of $A$ and $B$) on $\Gamma_1$. The line $MA$ meets $\Gamma_2$ again at $P$. The line $MK$ meets $\Gamma_1$ again at $C$. The line $CA$ meets $\Gamma_2 $ again at $Q$. Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$.
[color=red][Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .][/color]
Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$.
[b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$.
[b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.
Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$.
(a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point.
(b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.
Let $O$ be the center of a circle. Let $OU,OV$ be perpendicular radii of the circle. The chord $PQ$ passes through the midpoint $M$ of $UV$. Let $W$ be a point such that $PM = PW$, where $U, V,M,W$ are collinear. Let $R$ be a point such that $PR = MQ$, where $R$ lies on the line $PW$. Prove that $MR = UV$.
[u]Alternative version:[/u] A circle $S$ is given with center $O$ and radius $r$. Let $M$ be a point whose distance from $O$ is $\frac{r}{\sqrt{2}}$. Let $PMQ$ be a chord of $S$. The point $N$ is defined by $\overrightarrow{PN} =\overrightarrow{MQ}$. Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$. Prove that $MR =\sqrt{2}r$.
Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.
(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass
through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass
through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).
Fix a point $A$, a circle $\Omega$ centered at $O$, and reals $r$ and $\theta$. Let $X$ and $Y$ be variable points on $\Omega$ so that $\measuredangle XOY = \theta$. The tangents to $\Omega$ at $X$ and $Y$ meet at $T$, and a dilation at $T$ with scale factor $r$ sends $A$ to $A'$. Let $P$ be the foot from $A'$ to $TX$.
$ $ $ $ $ $ $ $ $ $ Suppose that some point $P^*$ is the same for two different $X$. Show that $\measuredangle TXY = \measuredangle AP^\ast O$. (All angles are directed.)
Proposed by [i]Karn Chutinan[/i]
Let $\omega$ be a circle with center $O$ and a non-diameter chord $BC$ of $\omega$. A point $A$ varies on $\omega$ such that $\angle BAC<90^{\circ}$. Let $S$ be the reflection of $O$ through $BC$. Let $T$ be a point on $OS$ such that the bisector of $\angle BAC$ also bisects $\angle TAS$.
1. Prove that $TB=TC=TO$.
2. $TB, TC$ cut $\omega$ the second times at points $E, F$, respectively. $AE, AF$ cut $BC$ at $M, N$, respectively. Let $SM$ intersects the tangent line at $C$ of $\omega$ at $X$, $SN$ intersects the tangent line at $B$ of $\omega$ at $Y$. Prove that the bisector of $\angle BAC$ also bisects $\angle XAY$.
Let $n\geq 3$ be integer. Given a regular $n$-polygon $P$ with side length 4 on the plane $z=0$ in the $xyz$-space.Llet $G$ be a circumcenter of $P$. When the center of the sphere $B$ with radius 1 travels round along the sides of $P$, denote by $K_n$ the solid swept by $B$.
Answer the following questions.
(1) Take two adjacent vertices $P_1,\ P_2$ of $P$. Let $Q$ be the intersection point between the perpendicular dawn from $G$ to $P_1P_2$,
prove that $GQ>1$.
(2) (i) Express the area of cross section $S(t)$ in terms of $t,\ n$ when $K_n$ is cut by the plane $z=t\ (-1\leq t\leq 1)$.
(ii) Express the volume $V(n)$ of $K_n$ in terms of $n$.
(3) Denote by $l$ the line which passes through $G$ and perpendicular to the plane $z=0$. Express the volume $W(n)$ of the solid by generated by a rotation of $K_n$ around $l$ in terms of $n$.
(4) Find $\lim_{n\to\infty} \frac{V(n)}{W(n)} .$
We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly two units. Determine the number of regular size tables $n\times n$ ($n> 1$ - given a fixed positive integer). (We can assume that the rows and columns of the tables are numbered: the cases of coincidence in turn, reflect, and so considered different).
Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$.
Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.
Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$.
Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.