Found problems: 185
2017 Hanoi Open Mathematics Competitions, 13
Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?
2010 Thailand Mathematical Olympiad, 4
For $i = 1, 2$ let $\vartriangle A_iB_iC_i$ be a triangle with side lengths $a_i, b_i, c_i$ and altitude lengths $p_i, q_i, r_i$.
Define $a_3 =\sqrt{a_1^2 + a_2^2}, b_3 =\sqrt{b_1^2 + b_2^2}$ , and $c_3 =\sqrt{c_1^2 + c_2^2}$.
Prove that $a_3, b_3, c_3$ are side lengths of a triangle, and if $p_3, q_3, r_3$ are the lengths of altitudes of this triangle,
then $p_3^2 \ge p_1^2 +p_2^2$, $q_3^2 \ge q_1^2 +q_2^2$ , and $r_3^2 \ge r_1^2 +r_2^2$
1983 Poland - Second Round, 5
The bisectors of the angles $ CAB, ABC, BCA $ of the triangle $ ABC $ intersect the circle circumcribed around this triangle at points $ K, L, M $, respectively. Prove that $$ AK+BL+CM > AB+BC+CA.$$
2001 Switzerland Team Selection Test, 2
If $a,b$, and $c$ are the sides of a triangle, prove the inequality $\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}$.
When does equality occur?
2007 Thailand Mathematical Olympiad, 6
A triangle has perimeter $2s$, inradius $r$, and incenter $I$. If $s_a, s_b$ and $s_c$ are the distances from $I$ to the three vertices, then show that $$\frac34 +\frac{r}{s_a}+\frac{r}{s_b}+\frac{r}{s_c} \le \frac{s^2}{12r^2}$$
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.
1978 Kurschak Competition, 3
A triangle has inradius $r$ and circumradius $R$. Its longest altitude has length $H$. Show that if the triangle does not have an obtuse angle, then $H \ge r+R$. When does equality hold?
2009 Postal Coaching, 1
In a triangle $ABC$, let $D,E, F$ be interior points of sides $BC,CA,AB$ respectively. Let $AD,BE,CF$ meet the circumcircle of triangle $ABC$ in $K, L,M$ respectively. Prove that $\frac{AD}{DK} + \frac{BE}{EL} + \frac{CF}{FM} \ge 9$. When does the equality hold?
1999 French Mathematical Olympiad, Problem 3
For which acute-angled triangles is the ratio of the smallest side to the inradius the maximum?
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 $kV^2$.