Found problems: 649
Novosibirsk Oral Geo Oly VII, 2023.5
One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?
2019 BMT Spring, 16
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 $A$.
2021 Estonia Team Selection Test, 3
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
2017 Swedish Mathematical Competition, 3
Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that $2 | XY | \le | AD | + | BC |$. When is equality achieved?
1988 IMO Longlists, 4
The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$
1981 IMO, 1
Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]
2022 IFYM, Sozopol, 3
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.
2016 India IMO Training Camp, 1
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.
Ukraine Correspondence MO - geometry, 2003.5
Let $O$ be the center of the circle $\omega$, and let $A$ be a point inside this circle, different from $O$. Find all points $P$ on the circle $\omega$ for which the angle $\angle OPA$ acquires the greatest value.
1996 Estonia Team Selection Test, 2
Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$
2021 239 Open Mathematical Olympiad, 5
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 $ABC$.
2001 Tuymaada Olympiad, 6
On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$. Prove that $PQ \le \frac{1}{2} AB$.
VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.
2024 pOMA, 3
Let $ABC$ be a triangle with circumcircle $\Omega$, and let $P$ be a point on the arc $BC$ of $\Omega$ not containing $A$. Let $\omega_B$ and $\omega_C$ be circles respectively passing through $B$ and $C$ and such that both of them are tangent to line $AP$ at point $P$. Let $R$, $R_B$, $R_C$ be the radii of $\Omega$, $\omega_B$, and $\omega_C$, respectively.
Prove that if $h$ is the distance from $A$ to line $BC$, then
\[
\frac{R_B+R_C}{R} \le \frac{BC}{h}.
\]
1978 Chisinau City MO, 157
Prove that the side $AB$ of a convex quadrilateral $ABCD$ is less than its diagonal $AC$ if $|AB|+|BC| \le |AC| +| CD|$.
1990 All Soviet Union Mathematical Olympiad, 532
If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.
1989 IMO Longlists, 14
For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality
\[ 16Q^3 \geq 27 r^4 P,\]
where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.
2015 Balkan MO Shortlist, A2
Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that:
$$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c}
\leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$
(Albania)
2018 Turkey MO (2nd Round), 4
In a triangle $ABC$, the bisector of the angle $A$ intersects the excircle that is tangential to side $[BC]$ at two points $D$ and $E$ such that $D\in [AE]$. Prove that,
$$
\frac{|AD|}{|AE|}\leq \frac{|BC|^2}{|DE|^2}.
$$
2020 Princeton University Math Competition, B3
Let $ABC$ be a triangle and let the points $D, E$ be on the rays $AB$, $AC$ such that $BCED$ is cyclic. Prove that the following two statements are equivalent:
$\bullet$ There is a point $X$ on the circumcircle of $ABC$ such that $BDX$, $CEX$ are tangent to each other.
$\bullet$ $AB \cdot AD \le 4R^2$, where $R$ is the radius of the circumcircle of $ABC$.
1989 French Mathematical Olympiad, Problem 3
Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$, the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$.
VMEO III 2006 Shortlist, G3
The tetrahedron $OABC$ has all angles at vertex $O$ equal to $60^o$. Prove that $$AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2$$
Ukrainian TYM Qualifying - geometry, IV.10
Given a triangle $ABC$ and points $D, E, F$, which are points of contact of the inscribed circle to the sides of the triangle.
i) Prove that $\frac{2pr}{R} \le DE + EF + DF \le p$
($p$ is the semiperimeter, $r$ and $R$ are respectively the radius of the inscribed and circumscribed circle of $\vartriangle ABC$).
ii). Find out when equality is achieved.
1988 Czech And Slovak Olympiad IIIA, 6
Inside the triangle $A_1A_2A_3$ with sides $a_1$, $a_2$, $a_3$, three points are given, which we label $P_1$, $P_2$, $P_3$ so that the product of their distances from the corresponding sides $a_1$, $a_2$, $a_3$ is as large as possible. Prove that the triangles $P_1A_2A_3$, $A_1P_2A_3$, $A_1A_2P_3$ cover the triangle.
[hide=original wording]V trojúhelníku A1A2A3 se stranami a1, a2, a3 jsou dány tři body, které označíme Pi, P2, P3 tak, aby součin jejich vzdáleností od odpovídajících stran a1, a2, a3 byl co největší. Dokažte, že trojúhelníky P1A2A3, A1P2A3, A1A2P3 pokrývají trojúhelník.[/quote]