Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

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 $1 \leq i < j < k \leq n$.

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

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.

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? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

Consider the functions $ f(x) = ax^{2} + bx + c $ , $ g(x) = cx^{2} + bx + a $, where a, b, c are real numbers. Given that $ |f(-1)| \leq 1 $, $ |f(0)| \leq 1 $, $ |f(1)| \leq 1 $, prove that $ |f(x)| \leq \frac{5}{4} $ and $ |g(x)|  \leq 2 $ for $ -1 \leq  x \leq 1 $.

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]

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain.

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?

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.

Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that \[ AB\plus{}BC\plus{}CA \plus{} DE < AD \plus{} AE \plus{} BD\plus{}BE \plus{} CD\plus{}CE .\]

We say that a polygon $P$ is [i]inscribed[/i] in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is [i]circumscribed[/i] to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$. Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.

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

The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?

Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$? $ \textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256} $

The sides of a triangle with positive area have lengths $ 4$, $ 6$, and $ x$. The sides of a second triangle with positive area have lengths $ 4$, $ 6$, and $ y$. What is the smallest positive number that is [b]not[/b] a possible value of $ |x \minus{} y|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10$

Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$. Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.

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

Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron? (S. Markelov)

Let $a$ be a real number. 1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$. 2) Prove that the area of this triangle does not depend on $a$.

Let $S=\{1,2,\ldots,10\}$. Three numbers are chosen with replacement from $S$. If the chosen numbers denote the lengths of sides of a triangle, then the probability that they will form a triangle is: $\textbf{(A)}~\frac{101}{200}$ $\textbf{(B)}~\frac{99}{200}$ $\textbf{(C)}~\frac12$ $\textbf{(D)}~\frac{110}{200}$

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

Let $\alpha,\beta,\gamma$ measures of angles of opposite to the sides of triangle with measures $a,b,c$ respectively.Prove that $$2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}$$