Found problems: 649
In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2 <r_a$ .
(here $r$ is the radius of the circle inscribed in the triangle $ABC$, $r_a$ is the radius of an exscribed circle that touches the sides of $BC$).
(Mykola Moroz)
The median $AD$ is drawn in triangle $ABC$. Point $E$ is selected on segment $AC$, and on the ray $DE$ there is a point $F$, and $\angle ABC = \angle AED$ and $AF // BC$. Prove that from segments $BD, DF$ and $AF$, you can make a triangle, the area of which is not less half the area of triangle $ABC$.
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 $ A_1A_2A_3A_4$ be a tetrahedron, $ G$ its centroid, and $ A'_1, A'_2, A'_3,$ and $ A'_4$ the points where the circumsphere of $ A_1A_2A_3A_4$ intersects $ GA_1,GA_2,GA_3,$ and $ GA_4,$ respectively. Prove that
\[ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4\]
and
\[ \frac{1}{GA'_1} \plus{} \frac{1}{GA'_2} \plus{} \frac{1}{GA'_3} \plus{} \frac{1}{GA'_4} \leq \frac{1}{GA_1} \plus{} \frac{1}{GA_2} \plus{} \frac{1}{GA_3} \plus{} \frac{1}{GA_4}.\]
Let $ABC$ be an acute-angled triangle with altitudes $AD,BE,$ and $CF$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that \[\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.\]
For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)
Let $ABC$ be an acute triangle, with median $AM$, height $AH$ and internal angle bisector $AL$. Suppose that $B, H, L, M, C$ are collinear in that order, and $LH<LM$. Prove that $BC>2AL$.
Given a triangle $ABC$, inside which the point $M$ is marked. On the sides $BC,CA$ and $AB$ the following points $A_1,B_1$ and $C_1$ are chosen, respectively, that $MA_1 \parallel CA$, $MB_1 \parallel AB$, $MC_1 \parallel BC$. Let S be the area of triangle $ABC, Q_M$ be the area of the triangle $A_1 B_1 C_1$.
a) Prove that if the triangle $ABC$ is acute, and M is the point of intersection of its altitudes , then $3Q_M \le S$. Is there such a number $k> 0$ that for any acute-angled triangle $ABC$ and the point $M$ of intersection of its altitudes, such thatthe inequality $Q_M> k S$ holds?
b) For cases where the point $M$ is the point of intersection of the medians, the center of the inscribed circle, the center of the circumcircle, find the largest $k_1> 0$ and the smallest $k_2> 0$ such that for an arbitrary triangle $ABC$, holds the inequality $k_1S \le Q_M\le k_2S$ (for the center of the circumscribed circle, only acute-angled triangles $ABC$ are considered).
Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality
\[AF + AH + AC \leq AB + AD + AE + AG.\]
In what cases does equality hold?
[i]Proposed by France.[/i]
Prove that for any interior point of a triangle the sum of squares of distances from it to the sides of a triangle is not less than $\frac{4S^2}{9R^2}$.
[hide=about S,R]they are not defined, but I suppose they mean it's area and circumradii respectively[/hide]
Prove theat for an arbitrary triangle holds the inequality $$a \cos A+ b \cos B + c \cos C \le p ,$$ where $a, b, c$ are the sides of the triangle, $A, B, C$ are the angles, $p$ is the semiperimeter.
Prove that in a triangle the following inequality holds: $$s\sqrt3 \ge \ell_a + \ell_b + \ell_c$$ where $\ell_a$ is the length of the angle bisector from angle $A$, and $s$ is the semiperimeter of the triangle
Let $ABCD$ be a quadrilateral with sides $AB$, $BC$, $CD$, $DA$ and diagonals $AC$, $BD$. Suppose that all sides of the quadrilateral have length greater than $ 1$, and that the difference between any side and diagonal is less than 1. Prove that the following inequality holds $$(AB + BC + CD + DA + AC + BD)^2 > 2|AC^3 - BC^3| + 2|BD^3 - AD^3| - (AB + CD)^3$$
A $ABC$ triangle divide by a $d$ line such that, new two pieces' areas are equal. $d$ line intersects with $[AB]$ at $D$, $[AC]$ at $E$. Prove that
$$\frac{AD+AE}{BD+DE+EC+CB} > \frac{1}{4}$$
Quadrilateral $ABCD$ is circumscribed around circle $k$. Gind the smallest possible value of
$$\frac{AB + BC + CD + DA}{AC + BD}$$, as well as all quadrilaterals with the above property where it is reached.
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]
Hiven an acute triangle $ABC$, consider the midpoints $M$ and $N$ of the sides $AB$ and $AC$, respectively. If point $S$ is variable on side $BC$, prove that $$(MB - MS)(NC - NS) \le 0$$
Two straight lines are given on a plane, intersecting at point $O$ at an angle $a$. Let $A$, $B$ and $C $ be three points on one of the lines, located on one side of$ O$ and following in the indicated order, $M$ be an arbitrary point on another line, different from $O$, Let $\angle AMB=\gamma$, $\angle BMC = \phi$. Consider the function $F(M) = ctg \gamma + ctg \phi$ . Prove that$ F(M)$ takes the smallest value on each of the rays into which $O$ divides the second straight line. (Each has its own.) Let us denote one of these smallest values by $q$, and the other by $p$. Prove that the exprseeion $\frac{p}{q}$ is independent of choice of points $A$, $B$ and $C$. Express this relationship in terms of $a$.
Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$.
The lengths, $a$ and $b$, of two sides of a triangle are known.
(a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value?
(b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?
Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$
If $a, b, c$ represent the lengths of the sides of a triangle, prove that:
$$\frac{a}{b-a+c}+ \frac{b}{b-a+c}+ \frac{c}{b-a+c} \ge 3$$
Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$, $K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$.
I. Voronovich
A point $P$ lies inside $\vartriangle ABC$ such that the values of areas of $\vartriangle PAB, \vartriangle PBC, \vartriangle PCA$ can form a triangle. Let $BC = a,CA = b,AB = c, PA = x,PB = y, PC = z$, prove that
$$\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z} \le a + b + c$$
Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.