Found problems: 649
1989 IMO Longlists, 40
Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that:
\[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}}
\]
1999 IMO Shortlist, 1
Let ABC be a triangle and $M$ be an interior point. Prove that
\[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]
2011 Saudi Arabia Pre-TST, 1
Let $ABC$ be a triangle with $\angle A = 90^o$ and let $P$ be a point on the hypotenuse $BC$. Prove that
$$\frac{AB^2}{PC}+\frac{AC^2}{PB} \ge \frac{BC^3}{PA^2 + PB \cdot PC}$$
1970 IMO Longlists, 17
In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?
Indonesia MO Shortlist - geometry, g8
Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that
$$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$
2008 Mathcenter Contest, 3
Let $ABC$ be a triangle whose side lengths are opposite the angle $A,B,C$ are $a,b,c$ respectively. Prove that $$\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}$$.
[i](nooonuii)[/i]
1998 Yugoslav Team Selection Test, Problem 2
In a convex quadrilateral $ABCD$, the diagonal $AC$ intersects the diagonal $BD$ at its midpoint $S$. The radii of incircles of triangles $ABS,BCS,CDS,DAS$ are $r_1,r_2,r_3,r_4$, respectively. Prove that
$$|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.$$
1971 Spain Mathematical Olympiad, 4
Prove that in every triangle with sides $a, b, c$ and opposite angles $A, B, C$, is fulfilled (measuring the angles in radians) $$\frac{a A+bB+cC}{a+b+c} \ge \frac{\pi}{3}$$
Hint: Use $a \ge b \ge c \Rightarrow A \ge B \ge C$.
1988 Dutch Mathematical Olympiad, 4
Given is an isosceles triangle $ABC$ with $AB = 2$ and $AC = BC = 3$. We consider squares where $A, B$ and $C$ lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.
IV Soros Olympiad 1997 - 98 (Russia), 11.4
Find the largest value of the area of the projection of the cylinder onto the plane if its radius is $r$ and its height is $h$ (orthogonal projection).
2013 JBMO Shortlist, 5
A circle passing through the midpoint $M$ of the side $BC$ and the vertex $A$ of the triangle $ABC$ intersects the segments $AB$ and $AC$ for the second time in the points $P$ and $Q$, respectively. Prove that if $\angle BAC=60^{\circ}$, then $AP+AQ+PQ<AB+AC+\frac{1}{2} BC$.
1948 Moscow Mathematical Olympiad, 149
Let $R$ and $r$ be the radii of the circles circumscribed and inscribed, respectively, in a triangle. Prove that $R \ge 2r$, and that $R = 2r$ only for an equilateral triangle.
1989 IMO Shortlist, 20
Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that
[b]i.)[/b] no three points of $ S$ are collinear, and
[b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$
Prove that:
\[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n}
\]
2023 Cono Sur Olympiad, 3
In a half-plane, bounded by a line \(r\), equilateral triangles \(S_1, S_2, \ldots, S_n\) are placed, each with one side parallel to \(r\), and their opposite vertex is the point of the triangle farthest from \(r\).
For each triangle \(S_i\), let \(T_i\) be its medial triangle. Let \(S\) be the region covered by triangles \(S_1, S_2, \ldots, S_n\), and let \(T\) be the region covered by triangles \(T_1, T_2, \ldots, T_n\).
Prove that \[\text{area}(S) \leq 4 \cdot \text{area}(T).\]
1967 IMO Shortlist, 1
Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$
1989 IMO, 4
Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that:
\[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}}
\]
1988 Greece National Olympiad, 2
Let $ABC$ be a triangle inscribed in circle $C(O,R)$. Let $M$ ba apoint on the arc $BC$ . Let $D,E,Z$ be the feet of the perpendiculars drawn from $M$ on lines $AB,AC,BC$ respectively. Prove that $\frac{(BC)^2}{(MZ)^2} \ge 8\frac{R U_a}{(MD)\cdot(ME)}$ where $U_a$ is the altitude drawn on $BC$.
1985 IMO Longlists, 77
Two equilateral triangles are inscribed in a circle with radius $r$. Let $A$ be the area of the set consisting of all points interior to both triangles. Prove that $2A \geq r^2 \sqrt 3.$
2019 Saint Petersburg Mathematical Olympiad, 3
Prove that the distance between the midpoint of side $BC$ of triangle $ABC$ and the midpoint of the arc $ABC$ of its circumscribed circle is not less than $AB / 2$
Novosibirsk Oral Geo Oly VII, 2019.6
Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.
2019 BMT Spring, 14
A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let $a$ be the distance that the laser travels. What is the smallest possible value of $a^2$ such that $a > 2019$?
You need not simplify/compute exponents.
2000 Belarus Team Selection Test, 4.2
Let ABC be a triangle and $M$ be an interior point. Prove that
\[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]
2015 Indonesia MO Shortlist, G1
Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.
2014 Turkey MO (2nd round), 3
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation
\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]
holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.
VII Soros Olympiad 2000 - 01, 10.8
There is a set of triangles, in each of which the smallest angle does not exceed $36^o$ . A new one is formed from these triangles according to the following rule: the smallest side of the new one is equal to the sum of the smallest sides of these triangles, its middle side is equal to the sum of the middle sides, and the largest is the sum of the largest ones. Prove that the sine of the smallest angle of the resulting triangle is less than $2 \sin 18^o$ .