Found problems: 649
Kyiv City MO Juniors Round2 2010+ geometry, 2021.7.41
Point $C$ lies inside the right angle $AOB$. Prove that the perimeter of triangle $ABC$ is greater than $2 OC$.
Kyiv City MO Juniors 2003+ geometry, 2012.7.4
Given an isosceles triangle $ABC$ with a vertex at the point $B$. Based on $AC$, an arbitrary point $D $ is selected, different from the vertices $A$ and $C $. On the line $AC $ select the point $E $ outside the segment $AC$, for which $AE = CD$. Prove that the perimeter $\Delta BDE$ is larger than the perimeter $\Delta ABC$.
1995 Tournament Of Towns, (479) 3
A rectangle with sides of lengths $a$ and $b$ ($a > b$) is cut into rightangled triangles so that any two of these triangles either have a common side, a common vertex or no common points. Moreover, any common side of two triangles is a leg of one of them and the hypotenuse of the other. Prove that $a > 2b$.
(A Shapovalov)
Novosibirsk Oral Geo Oly IX, 2019.7
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 $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$
2020 Swedish Mathematical Competition, 2
The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$.
2016 Peru Cono Sur TST, P2
Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$
1969 All Soviet Union Mathematical Olympiad, 127
Let $h_k$ be an apothem of the regular $k$-gon inscribed into a circle with radius $R$. Prove that $$(n + 1)h_{n+1} - nh_n > R$$
1967 IMO Longlists, 23
Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality
\[af^2 + bfg +cg^2 \geq 0\]
holds if and only if the following conditions are fulfilled:
\[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]
1986 IMO Longlists, 70
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.
1995 Tuymaada Olympiad, 8
Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.
2010 Balkan MO Shortlist, G2
Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle
2017 Caucasus Mathematical Olympiad, 7
$8$ ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding $1$.
2017 Germany, Landesrunde - Grade 11/12, 5
In a right-angled triangle let $r$ be the inradius and $s_a,s_b$ be the lengths of the medians of the legs $a,b$.
Prove the inequality \[ \frac{r^2}{s_a^2+s_b^2} \leq \frac{3-2 \sqrt2}{5}. \]
Kyiv City MO 1984-93 - geometry, 1984.10.5
The vertices of a regular hexagon $A_1,A_2,...,A_6$ lie respectively on the sides $B_1B_2$, $B_2B_3$, $B_3B_4$, $B_4B_5$, $B_5B_6$, $B_6B_1$ of a convex hexagon $B_1B_2B_3B_4B_5B_6$. Prove that
$$S_{B_1B_2B_3B_4B_5B_6} \le \frac32 S_{A_1A_2A_3A_4A_5A_6}.$$
2019 Mediterranean Mathematics Olympiad, 4
Let $P$ be a point in the interior of an equilateral triangle with height $1$, and let $x,y,z$ denote the distances from $P$ to the three sides of the triangle. Prove that
\[ x^2+y^2+z^2 ~\ge~ x^3+y^3+z^3 +6xyz \]
1963 IMO Shortlist, 3
In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation
\[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \]
Prove that $a_{1}=a_{2}= \ldots= a_{n}$.
2006 Spain Mathematical Olympiad, 3
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?
2025 Sharygin Geometry Olympiad, 11
A point $X$ is the origin of three rays such that the angle between any two of them equals $120^{\circ}$. Let $\omega$ be an arbitrary circle with radius $R$ such that $X$ lies inside it, and $A$, $B$, $C$ be the common points of the rays with this circle. Find $max(XA+XB+XC)$.
Proposed by: F.Nilov
2013 Bogdan Stan, 1
$ M,N,P,Q,R,S $ are the midpoints of the sides $ AB,BC,CD,DE,EF,FA $ of a convex hexagon $ ABCDEF. $
[b]a)[/b] Show that with the segments $ MQ,NR,PS, $ it can be formed a triangle.
[b]b)[/b] Show that a triangle formed with the segments $ MQ,NR,PS $ is right if and only if ether $ MQ\perp NR $ or $ MQ\perp PS $ or $ PS\perp RN. $
[i]Vasile Pop[/i]
2000 Rioplatense Mathematical Olympiad, Level 3, 2
In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $$\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$$
Note: $(XYZ)$ is the area of triangle $XYZ$.
Kyiv City MO 1984-93 - geometry, 1993.10.4
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.
1988 ITAMO, 6
The edge lengths of the base of a tetrahedron are $a,b,c$, and the lateral edge lengths are $x,y,z$. If $d$ is the distance from the top vertex to the centroid of the base, prove that $x+y+z \le a+b+c+3d$.
1998 Tuymaada Olympiad, 3
The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where $r$ is the radius of the inscribed circle of the triangle.
1983 All Soviet Union Mathematical Olympiad, 365
One side of the rectangle is $1$ cm. It is known that the rectangle can be divided by two orthogonal lines onto four rectangles, and each of the smaller rectangles has the area not less than $1$ square centimetre, and one of them is not less than $2$ square centimetres. What is the least possible length of another side of big rectangle?
2009 Ukraine Team Selection Test, 12
Denote an acute-angle $\vartriangle ABC $ with sides $a, b, c $ respectively by ${{H}_{a}}, {{H}_{b}}, {{H}_{c}} $ the feet of altitudes ${{h}_{a}}, {{h}_{b}}, {{h}_{c}} $. Prove the inequality:
$$\frac {h_ {a} ^{2}} {{{a} ^{2}} - CH_ {a} ^{2}} + \frac{h_{b} ^{2}} {{{ b}^{2}} - AH_{b} ^{2}} + \frac{h_{c}^{2}}{{{c}^{2}} - BH_{c}^{2}} \ge 3 $$
(Dmitry Petrovsky)