Found problems: 185
1976 Czech and Slovak Olympiad III A, 5
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 $o_1\le o_2.$
2012 Bogdan Stan, 4
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:
$$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$
2001 Estonia Team Selection Test, 2
Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
1971 Bulgaria National Olympiad, Problem 6
In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that
(a) $m^2+n^2+p^2\ge18r^2$;
(b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$.
1972 Polish MO Finals, 4
Points $A$ and $B$ are given on a line having no common points with a sphere $K$. The feet $P$ of the perpendicular from the center of $K$ to the line $AB$ is positioned between $A$ and $B$, and the lengths of segments $AP$ and $BP$ both exceed the radius of $K$. Consider the set $Z$ of all triangles $ABC$ whose sides $AC$ and $BC$ are tangent to $K$. Prove that among all triangles in $Z$, a triangle $T$ with a maximum perimeter also has a maximum area.
1963 Kurschak Competition, 3
A triangle has no angle greater than $90^o$. Show that the sum of the medians is greater than four times the circumradius.
1942 Eotvos Mathematical Competition, 1
Prove that in any triangle, at most one side can be shorter than the altitude from the opposite vertex.
1952 Kurschak Competition, 3
$ABC$ 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 $ABC$.
1956 Polish MO Finals, 5
Prove that every polygon with perimeter $ 2a $ can be covered by a disk with diameter $ a $.
1974 Bulgaria National Olympiad, Problem 6
In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true
$$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$
where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$.
[i]V. Petkov[/i]
1964 Swedish Mathematical Competition, 1
Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.
1995 Chile National Olympiad, 7
In a semicircle of radius $4$ three circles are inscribed, as indicated in the figure. Larger circles have radii $ R_1 $ and $ R_2 $, and the larger circle has radius $ r $.
a) Prove that $ \dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}} $
b) Prove that $ R_1 + R_2 \le 8 (\sqrt{2} -1) $
c) Prove that $ r \le \sqrt{2} -1 $
[img]https://cdn.artofproblemsolving.com/attachments/0/9/aaaa65d1f4da4883973751e1363df804b9944c.jpg[/img]
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.
2011 Indonesia TST, 1
Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that
$$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$
1978 Bulgaria National Olympiad, Problem 6
The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$.
[i]Jordan Tabov[/i]
2011 Indonesia TST, 1
Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that
$$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$
2010 Junior Balkan Team Selection Tests - Moldova, 6
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 $E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}$.
1963 Polish MO Finals, 3
From a given triangle, cut out the rectangle with the largest area.
1975 Polish MO Finals, 5
Show that it is possible to circumscribe a circle of radius $R$ about, and inscribe a circle of radius $r$ in some triangle with one angle equal to $a$, if and only if $$\frac{2R}{r} \ge \dfrac{1}{ \sin \frac{a}{2} \left(1- \sin \frac{a}{2} \right)}$$
2005 Chile National Olympiad, 1
In the center of the square of side $1$ shown in the figure is an ant. At one point the ant starts walking until it touches the left side $(a)$, then continues walking until it reaches the bottom side $(b)$, and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of $2$.
[asy]
unitsize(2 cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
label("$a$", (0,0.5), W);
label("$b$", (0.5,0), S);
dot((0.5,0.5));
[/asy]
1992 Czech And Slovak Olympiad IIIA, 2
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$
1995 Singapore MO Open, 3
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]
Indonesia MO Shortlist - geometry, g9
Given a triangle $ABC$, the points $D$, $E$, and $F$ lie on the sides $BC$, $CA$, and $AB$, respectively, are such that
$$DC + CE = EA + AF = FB + BD.$$ Prove that $$DE + EF + FD \ge \frac12 (AB + BC + CA).$$
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 $$\frac34 +\frac{r}{s_a}+\frac{r}{s_b}+\frac{r}{s_c} \le \frac{s^2}{12r^2}$$
1973 Poland - Second Round, 1
Prove that if positive numbers $ x, y, z $ satisfy the inequality
$$
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,$$
then they are the lengths of the sides of a certain triangle.