Olympiad problems

2013 AMC 12/AHSME, 24

Tags: probability , geometry , inequalities , trigonometry , triangle inequality , trig identities , law of sines

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? $ \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C)} \ \frac{111}{143} \qquad \textbf{(D)} \ \frac{81}{104} \qquad \textbf{(E)} \ \frac{223}{286}$

2001 National Olympiad First Round, 18

Tags: inequalities , triangle inequality , geometry , perpendicular bisector

A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

2009 Indonesia TST, 2

Tags: geometry , incenter , triangle inequality , geometric inequality , angle bisector

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

2005 National Olympiad First Round, 11

Tags: inequalities , analytic geometry , triangle inequality , geometry

For the real pairs $(x,y)$ satisfying the equation $x^2 + y^2 + 2x - 6y = 6$, which of the following cannot be equal to $(x-1)^2 + (y-2)^2$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 23 \qquad\textbf{(E)}\ 30 $

2006 Iran Team Selection Test, 4

Tags: induction , function , triangle inequality , inequalities

Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that \[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]

1983 AMC 12/AHSME, 29

Tags: analytic geometry , inequalities , triangle inequality

A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A$, $B$, $C$ and $D$. Also, let the distances from $P$ to $A$, $B$ and $C$, respectively, be $u$, $v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$? $ \textbf{(A)}\ 1 + \sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{2}\qquad\textbf{(C)}\ 2 + \sqrt{2}\qquad\textbf{(D)}\ 3\sqrt{2}\qquad\textbf{(E)}\ 3 + \sqrt{2}$

1990 India National Olympiad, 5

Tags: search , inequalities , triangle inequality , limit , algebra

Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity \[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\] must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?

2019 Tournament Of Towns, 2

Tags: geometry , geometric inequality , area of a triangle , triangle inequality , inequalities

Two acute triangles $ABC$ and $A_1B_1C_1$ are such that $B_1$ and $C_1$ lie on $BC$, and $A_1$ lies inside the triangle $ABC$. Let $S$ and $S_1$ be the areas of those triangles respectively. Prove that $\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}$ (Nairi Sedrakyan, Ilya Bogdanov)

2009 Purple Comet Problems, 8

Tags: calculus , integration , inequalities , triangle inequality

Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$.

2006 Tuymaada Olympiad, 2

Tags: inequalities , circumcircle , euler , trigonometry , triangle inequality , geometry

Let $ABC$ be a triangle, $G$ it`s centroid, $H$ it`s orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$). It is known that $MG=R$, where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$. [i]Proposed by F. Bakharev[/i]