Found problems: 649
2015 BMT Spring, 9
Find the side length of the largest square that can be inscribed in the unit cube.
Ukrainian TYM Qualifying - geometry, IV.7
Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.
2019 Polish Junior MO Finals, 4.
The point $D$ lies on the side $AB$ of the triangle $ABC$. Assume that there exists such a point $E$ on the side $CD$, that
$$
\sphericalangle EAD = \sphericalangle AED \quad \text{and} \quad \sphericalangle ECB = \sphericalangle CEB.
$$
Show that $AC + BC > AB + CE$.
Ukrainian TYM Qualifying - geometry, VI.18
The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices:
1) $T_3$ and $T_3 (A_2)$
2) $T_k$ and $T_k (A_1) $
3) $T_k$ and $T_{k+1}$
1966 IMO Shortlist, 5
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 $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$
2001 Chile National Olympiad, 3
In a triangle $ \vartriangle ABC $, let $ h_a, h_b $ and $ h_c $ the atlitudes. Let $ D $ be the point where the inner bisector of $ \angle BAC $ cuts to the side $ BC $ and $ d_a $ is the distance from the $ D $ point next to $ AB $. The distances $ d_b $ and $ d_c $ are similarly defined. Show that:
$$ \dfrac {3} {2} \le \dfrac {d_a} {h_a} + \dfrac {d_b} {h_b} + \dfrac {d_c} {h_c} $$
For what kind of triangles does the equality hold?
2001 239 Open Mathematical Olympiad, 7
The quadrangle $ ABCD $ contains two circles of radii $ R_1 $ and $ R_2 $ tangent externally. The first circle touches the sides of $ DA $,$ AB $ and $ BC $, moreover, the sides of $ AB $ at the point $ E $. The second circle touches sides $ BC $, $ CD $ and $ DA $, and sides $ CD $ at $ F $. Diagonals of the quadrangle intersect at $ O $. Prove that $ OE + OF \leq 2 (R_1 + R_2) $.
(F. Bakharev, S. Berlov)
1982 Austrian-Polish Competition, 8
Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$.
Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.
1966 IMO Shortlist, 63
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
2013 HMNT, 5
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Company $XYZ$ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$, $B$, and $C$. There are $1$, $5$, and $4$ workers at $A$, $B$, and $C$, respectively. Find the minimum possible total distance Company $XYZ$'s workers have to travel to get to $P$.
2015 China Western Mathematical Olympiad, 7
Let $a\in (0,1)$, $f(z)=z^2-z+a, z\in \mathbb{C}$. Prove the following statement holds:
For any complex number z with $|z| \geq 1$, there exists a complex number $z_0$ with $|z_0|=1$, such that $|f(z_0)| \leq |f(z)|$.
2013 BMT Spring, 6
Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Because Bubble Boy loves Bubble Girl, he wants to reach her as quickly as possible, but he needs to bring a gift; luckily, there are plenty of gifts along the $x$-axis. Assuming that Bubble Girl remains stationary, find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then reach Bubble Girl (the bubble is a solid boundary, and anything the bubble can touch, Bubble Boy can touch too)
2003 Gheorghe Vranceanu, 4
Let $ I $ be the incentre of $ ABC $ and $ D,E,F $ be the feet of the perpendiculars from $ I $ to $ BC,CA,AB, $ respectively. Show that
$$ \frac{AB}{DE} +\frac{BC}{EF} +\frac{CA}{FD}\ge 6. $$
1979 IMO Shortlist, 10
Show that for any vectors $a, b$ in Euclidean space,
\[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\]
Remark. Here $\times$ denotes the vector product.
2022 Iran MO (3rd Round), 3
We have $n\ge3$ points on the plane such that no three are collinear. Prove that it's possible to name them $P_1,P_2,\ldots,P_n$ such that for all $1<i<n$, the angle $\angle P_{i-1}P_iP_{i+1}$ is acute.
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.
Kyiv City MO 1984-93 - geometry, 1989.8.2
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 $$\frac{m_a}{h_a}+\frac{m_b}{h_b}+\frac{m_c}{h_c} <1+\frac{R}{r}.$$
2016 Indonesia TST, 2
Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds:
\[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]
MIPT student olimpiad autumn 2024, 4
The ellipsoid $E$ is contained in the simplex $S$, which is located in the unit ball
B space $R^n$. Prove that the sum of the principal semi-axes of the ellipsoid $E$ is no more than
units.
Ukrainian TYM Qualifying - geometry, VI.14
A quadrilateral whose perimeter is equal to $P$ is inscribed in a circle of radius $R$ and is circumscribed around a circle of radius $r$. Check whether the inequality $P\le \frac{r+\sqrt{r^2+4R^2}}{2}$ holds.
Try to find the corresponding inequalities for the $n$-gon ($n \ge 5$) inscribed in a circle of radius $R$ and circumscribed around a circle of radius $r$.
1999 Bundeswettbewerb Mathematik, 3
Let $P$ be a point inside a convex quadrilateral $ABCD$. Points $K,L,M,N$ are given on the sides $AB,BC,CD,DA$ respectively such that $PKBL$ and $PMDN$ are parallelograms. Let $S,S_1$, and $S_2$ be the areas of $ABCD, PNAK$, and $PLCM$. Prove that $\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}$.
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.$
1985 Tournament Of Towns, (085) 1
$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$.
Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$
(V. Prasolov )
2023 Iranian Geometry Olympiad, 4
Let $ABCD$ be a convex quadrilateral. Let $E$ be the intersection of its diagonals. Suppose that $CD = BC = BE$. Prove that $AD + DC\ge AB$.
[i]Proposed by Dominik Burek - Poland[/i]
2018 German National Olympiad, 4
a) Let $a,b$ and $c$ be side lengths of a triangle with perimeter $4$. Show that
\[a^2+b^2+c^2+abc<8.\]
b) Is there a real number $d<8$ such that for all triangles with perimeter $4$ we have
\[a^2+b^2+c^2+abc<d \quad\]
where $a,b$ and $c$ are the side lengths of the triangle?