Found problems: 649
Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that
\[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \]
[i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$.
Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$.
[i]Proposed by Dirk Laurie, South Africa[/i]
The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.
Let $ABC$ be a triangle with $AB = 3$, $BC = 5$, $AC = 7$, and let $ P$ be a point in its interior. If $G_A$, $G_B$, $G_C$ are the centroids of $\vartriangle PBC$, $\vartriangle PAC$, $\vartriangle PAB$, respectively, find the maximum possible area of $\vartriangle G_AG_BG_C$.
Each side of a convex quadrilateral is less than $20$ cm. Prove that you can specify the vertex of the quadrilateral, the distance from which to any point $Q$ inside the quadrilateral is less than $15$ cm.
Let $A$ and $B$ be two given points, distance $1 $ apart. Determine a point $P$ on the line $AB$ such that
$$\frac{1}{1 + AP}+\frac{1}{1 + BP}$$ is a maximum.
Let $h_a,h_b,h_c$ be the altitudes, and let $m_a,m_b,m_c$ be the medians of the acute triangle drawn to the sides $a, b, c$ respectively. Let $r$ and $R$ be the radii of the inscribed and circumscribed circles. Prove that $$\frac{m_a}{h_a}+\frac{m_b}{h_b}+\frac{m_c}{h_c} <1+\frac{R}{r}.$$
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
In the square $ABCD$ we consider $ E \in (AB)$, $ F \in (AD)$ and $EF \cap AC = \{P\}$. Show that:
a) $\frac{1}{AE} + \frac{1}{AF} = \frac{\sqrt2}{AP}$
b) $AP^2 \le \frac{AE \cdot AF}{2}$
Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.
Let $S$ be a finite set of points in the plane,situated in general position(any three points in $S$ are not collinear),and let $$D(S,r)=\{\{x,y\}:x,y\in S,\text{dist}(x,y)=r\},$$ where $R$ is a positive real number,and $\text{dist}(x,y)$ is the euclidean distance between points $x$ and $y$.Prove that $$\sum_{r>0}|D(S,r)|^2\le\frac{3|S|^2(|S|-1)}{4}.$$
Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.
Let two circles $A$ and $B$ with unequal radii $r$ and $R$, respectively, be tangent internally at the point $A_0$. If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$, and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$, prove that
\[\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.\]
A convex quadrilateral with area $1$ is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area $< \frac38$.
The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon.
(a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.)
(b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter.
(c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths.
It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle.
(d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon
can be increased without changing the perimeter.
(e) Prove that the polygon of maximum area must be equiangular, or have all angles equal.
(f ) Prove that the polygon of maximum area is a regular polygon.
PS. You had better use hide for answers.
Medians $AD, BE$, and $CF$ of triangle $ABC$ intersect at point $M$. Is it possible that the circles with radii $MD, ME$, and $MF$
a) all have areas smaller than the area of triangle $ABC$,
b) all have areas greater than the area of triangle $ABC$,
c) all have areas equal to the area of triangle $ABC$?
In the triangle $ABC$, $\angle{B} = 120^{\circ}$, point $M$ is the midpoint of side $AC$. On the sides $AB$ and $BC$, the points $K$ and $L$ are chosen such that $KL \parallel AC$. Prove that $MK + ML \geq MA$.
Prove the inequality
\[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\]
for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$
Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$.
Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$
Prove that for each $\Delta ABC$ with an acute $\angle C$ the following inequality is true:
$(a^2+b^2) cos(\alpha -\beta )\leq 2ab$.
Let $ABCD$ be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $(AD-BC)(AB- CD) > 0$.
(V. Yasinsky)
Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
On the side $AD$ of the convex quadrilateral $ABCD$ with an acute angle at $B$, a point $E$ is marked.
It is known that $\angle CAD = \angle ADC=\angle ABE =\angle DBE$.
(Grade 9 version) Prove that $BE+CE<AD$.
(Grade 10 version) Prove that $\triangle BCE$ is isosceles.(Here the condition that $\angle B$ is acute is not necessary.)