Olympiad problems

2011 Indonesia TST, 1

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

Indonesia Regional MO OSP SMA - geometry, 2012.4

Tags: geometry , geometric inequality , geometric inequalities , trigonometric inequality , trigonometry

Given an acute triangle $ABC$. Point $H$ denotes the foot of the altitude drawn from $A$. Prove that $$AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC$$

2008 Postal Coaching, 2

Tags: perpendicular , distance , geometric inequalities , geometry

Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.

1988 Poland - Second Round, 3

Tags: geometry , geometric inequalities

Inside the acute-angled triangle $ ABC $ we consider the point $ P $ and its projections $ L, M, N $ to the sides $ BC, CA, AB $, respectively. Determine the point $ P $ for which the sum $ |BL|^2 + |CM|^2 + |AN|^2 $ is the smallest.

2020 Jozsef Wildt International Math Competition, W58

Tags: geometric inequalities , geometry , inequalities

In all triangles $ABC$ does it hold that: $$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

1990 Romania Team Selection Test, 5

Tags: geometry , circumcircle , circles , area , geometric inequalities , inequalities

Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$.

1976 Czech and Slovak Olympiad III A, 5

Tags: geometry , perimeter , geometric inequalities , convex , polygon , inequalities

Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$

1983 Swedish Mathematical Competition, 5

Tags: geometry , combinatorial geometry , combinatorics , disks , geometric inequalities , min , disc

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

1970 Poland - Second Round, 4

Tags: geometry , perimeter , geometric inequalities

Prove that if triangle $T_1$ contains triangle $T_2$, then the perimeter of triangle $T_1$ is not less than the perimeter of triangle $T_2$.

2015 Sharygin Geometry Olympiad, P15

Tags: geometry , geometric inequalities , inradius , circumradius

The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.

2013 239 Open Mathematical Olympiad, 7

Tags: geometry , geometric inequalities

Point $M$ is the midpoint of side $BC$ of convex quadrilateral $ABCD$. If $\angle{AMD} < 120^{\circ}$. Prove that $$(AB+AM)^2 + (CD+DM)^2 > AD \cdot BC + 2AB \cdot CD.$$

1915 Eotvos Mathematical Competition, 3

Tags: geometry , parallelogram , inscribed , geometric inequalities , area

Prove that a triangle inscribed in a parallelogram has at most half the area of the parallelogram.

2010 Junior Balkan Team Selection Tests - Moldova, 6

Tags: geometry , algebra , geometric inequalities , inequalities

In a right triangle with the length legs $b$ and $c$, and the length hypotenuse $a$, the ratio between the length of the hypotenuse and the length of the diameter of the inscribed circle does not exceed $1 + \sqrt2$. Determine the numerical value of the expression of $E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}$.

1988 Swedish Mathematical Competition, 1

Tags: geometric inequalities , altitude , geometry

Let $a > b > c$ be sides of a triangle and $h_a,h_b,h_c$ be the corresponding altitudes. Prove that $a+h_a > b+h_b > c+h_c$.

1972 Kurschak Competition, 1

Tags: geometry , geometric inequalities , inequalities

A triangle has side lengths $a, b, c$. Prove that $$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$

1968 Swedish Mathematical Competition, 3

Tags: geometry , geometric inequalities

Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?

2012 Bogdan Stan, 4

Tags: geometric inequalities , angle inequalities , geometry , rational geometry , ratio , inequalities

Let $ D $ be a point on the side $ BC $ (excluding its endpoints) of a triangle $ ABC $ with $ AB>AC, $ such that $ \frac{\angle BAD}{\angle BAC} $ is a rational number. Prove the following: $$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$

1952 Kurschak Competition, 3

Tags: geometry , perimeter , geometric inequalities , ratio

$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$, where $1/2 < k < 1$. Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$, and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$. Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$.

2015 Regional Olympiad of Mexico Center Zone, 6

Tags: geometry , geometric inequalities

We have $3$ circles such that any $2$ of them are externally tangent. Let $a$ be length of the outer tangent common to a pair of them. The lengths $b$ and $c$ are defined similarly. If $T$ is the sum of the areas of such circles, show that $\pi (a + b + c)^2 \le 12T $. Note: In In the case of externally tangent circles, the common external tangent is the segment tangent to them that touches them at different points.

2010 Estonia Team Selection Test, 3

Tags: inequalities , geometric inequalities , trigonometry , circumradius , geometry

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

2009 Postal Coaching, 3

Tags: inequalities , polygon , cyclic , geometric inequalities

Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.

1916 Eotvos Mathematical Competition, 2

Tags: geometry , geometric inequalities

Let the bisector of the angle at $C$ of triangle $ABC$ intersect side $AB$ in point $D$. Show that the segment $CD$ is shorter than the geometric mean of the sides $CA$ and $CB$. (The geometric mean of two positive numbers is the square root of their product; the geometric mean of $n$ numbers is the $n$-th root of their product.

2009 Estonia Team Selection Test, 4

Tags: geometry , geometric inequalities , inequalities , ratio

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

1995 Singapore MO Open, 3

Tags: geometric inequalities , geometry , perpendicular

Let $P$ be a point inside $\vartriangle ABC$. Let $D, E, F$ be the feet of the perpendiculars from $P$ to the lines $BC, CA$ and $AB$, respectively (see Fig. ). Show that (i) $EF = AP \sin A$, (ii) $PA+ PB + PC \ge 2(PE+ PD+ PF)$ [img]https://cdn.artofproblemsolving.com/attachments/d/f/f37d8764fc7d99c2c3f4d16f66223ef39dfd09.png[/img]

1928 Eotvos Mathematical Competition, 3

Tags: geometry , geometric inequalities , construction

Let $\ell$ be a given line, $A$ and $B$ given points of the plane. Choose a point $P$ on $\ell $ so that the longer of the segments $AP$, $BP$ is as short as possible. (If $AP = BP,$ either segment may be taken as the longer one.)