Olympiad problems

2013 AMC 12/AHSME, 24

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

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}$

2012 Balkan MO Shortlist, A1

Tags: trigonometry , triangle inequality , rearrangement inequality , function

Prove that \[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\] for all positive real numbers $x,y$ and $z$.

PEN R Problems, 7

Tags: limit , integration , 3d geometry , triangle inequality , geometry , calculus , inequalities

Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]

2006 AMC 10, 10

Tags: inequalities , triangle inequality , geometry , perimeter

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? $ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$

2006 Iran Team Selection Test, 4

Tags: function , triangle inequality , induction , inequalities

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

2014 Mediterranean Mathematics Olympiad, 1

Tags: induction , triangle inequality , n-variable inequality , inequalities , algebra

Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

1994 Romania TST for IMO, 4:

Tags: inequalities , triangle inequality , geometry

Let be given two concentric circles of radii $R$ and $R_1 > R$. Let quadrilateral $ABCD$ is inscribed in the smaller circle and let the rays $CD, DA, AB, BC$ meet the larger circle at $A_1, B_1, C_1, D_1$ respectively. Prove that $$ \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}$$ where $\sigma(P)$ denotes the area of a polygon $P.$

2005 China Team Selection Test, 3

Tags: algebra , triangle inequality , trigonometry , inequalities , polynomial , induction

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

2013 Kosovo National Mathematical Olympiad, 5

Tags: geometry , perimeter , triangle inequality , inequalities

Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle

1968 IMO Shortlist, 10

Tags: geometry , triangle inequality

Consider two segments of length $a, b \ (a > b)$ and a segment of length $c = \sqrt{ab}$. [b](a)[/b] For what values of $a/b$ can these segments be sides of a triangle ? [b](b)[/b] For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled ?

2004 IMC, 3

Tags: inequalities , triangle inequality

Let $D$ be the closed unit disk in the plane, and let $z_1,z_2,\ldots,z_n$ be fixed points in $D$. Prove that there exists a point $z$ in $D$ such that the sum of the distances from $z$ to each of the $n$ points is greater or equal than $n$.

2012 AMC 12/AHSME, 23

Tags: triangle inequality , algebra , polynomial , inequalities

Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a, b, c$ and $d$ are integers, $0 \le d \le c \le b \le a \le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1$. What is the sum of all values $P(1)$ over all the polynomials with these properties? $ \textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 120 $

1990 Baltic Way, 1

Tags: triangle inequality , inequalities

Numbers $1, 2, \dots , n$ are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?

1995 AMC 12/AHSME, 23

Tags: calculus , trigonometry , pythagorean theorem , geometry , triangle inequality , integration , inequalities

The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

2009 Indonesia TST, 2

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

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 \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

2012 China Team Selection Test, 1

Tags: triangle inequality , complex number , inequalities , algebra

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

2012 China Team Selection Test, 1

Tags: inequalities , triangle inequality , complex number , algebra

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

2011 IMO Shortlist, 5

Tags: triangle inequality , trigonometry , triangle , algebra

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

2006 Tuymaada Olympiad, 2

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

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]

2004 IMO, 4

Tags: triangle inequality , n-variable inequality , inequalities

Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

2008 AMC 12/AHSME, 19

Tags: triangle inequality , inequalities , function , complex number

A function $ f$ is defined by $ f(z) \equal{} (4 \plus{} i) z^2 \plus{} \alpha z \plus{} \gamma$ for all complex numbers $ z$, where $ \alpha$ and $ \gamma$ are complex numbers and $ i^2 \equal{} \minus{} 1$. Suppose that $ f(1)$ and $ f(i)$ are both real. What is the smallest possible value of $ | \alpha | \plus{} |\gamma |$? $ \textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad$

2014 India IMO Training Camp, 2

Tags: triangle inequality , inequalities , algebra

For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold: $x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$ $v_{1},v_{2},v_{3}$ satisfy triangle inequality $v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality. Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.

2015 Purple Comet Problems, 7

Tags: triangle inequality , inequalities , trooga inooga theorem

How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side lengths and perimeter less than 20?

1987 AIME Problems, 2

Tags: triangle inequality , inequalities , 3d geometry , sphere , geometry

What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$?

2009 National Olympiad First Round, 13

Tags: trapezoid , geometry , trigonometry , triangle inequality , inequalities

In trapezoid $ ABCD$, $ AB \parallel CD$, $ \angle CAB < 90^\circ$, $ AB \equal{} 5$, $ CD \equal{} 3$, $ AC \equal{} 15$. What are the sum of different integer values of possible $ BD$? $\textbf{(A)}\ 101 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 115 \qquad\textbf{(D)}\ 125 \qquad\textbf{(E)}\ \text{None}$