If a b c positive reals smaller than 1, prove: a+b+c+2abc>ab+bc+ca+2(abc)^(1/2)

$p, q, r$ are distinct prime numbers which satisfy $$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$ for natural number $A$. Find all values of $A$.

The sum of three consecutive integers is $15$. Determine their product.

For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. [list=a] [*] Prove that $S(n)\leq n^{2}-14$ for each $n\geq 4$. [*] Find an integer $n$ such that $S(n)=n^{2}-14$. [*] Prove that there are infinitely many integers $n$ such that $S(n)=n^{2}-14$. [/list]

If $8/19$ of the product of largest two elements of a positive integer set is not greater than the sum of other elements, what is the minimum possible value of the largest number in the set? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 19 \qquad\textbf{(E)}\ 20 $

Determine all positive integers $n$ for which $2^{n+1}-n^2$ is a prime number.

Find the sum of the indexes of the singular points other than zero of the vector field \[z\overline{z}^2+z^4+2\overline{z}^4\]

Let $ABCD$ be a square. Choose points $E$ on $BC$ and $F$ on $CD$ so that $\angle EAF=45^\circ$ and so that neither $E$ nor $F$ is a vertex of the square. The lines $AE$ and $AF$ intersect the circumcircle of the square in the points $G$ and $H$ distinct from $A$, respectively. Show that the lines $EF$ and $GH$ are parallel.

The natural numbers are colored green or blue so that: $\bullet$ The sum of a green and a blue is blue; $\bullet$ The product of a green and a blue is green. How many ways are there to color the natural numbers with these rules, so that $462$ are blue and $2016$ are green?

How many pairs of positive integers $(a,b)$ are there such that the roots of polynomial $x^2-ax-b$ are not greater than $5$? $ \textbf{(A)}\ 40 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 65 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ \text{None of the preceding} $

One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation: [b]Martian [/b]: I have spent $1/12$ of my life on Pluton. [b]Human [/b]: I also have. [b]Venusian [/b]: Me too. [b]Martian [/b]: But Venusian and I have spend much more time here than you, Human. [b]Human [/b]: That is true. However, Venusian and I are of the same age. [b]Venusian [/b]: Yes, I have lived $300$ Earth years. [b]Martian [/b]: Venusian and I have been on Pluton for the past $13$ years. It is known that Human and Martian together have lived $104$ Earth years. Find the ages of Martian, Venusian, and Human.* [hide="*"][i]*: Note that the numbers in the problem are not necessarily in base $10.$[/i][/hide]

On a straight line $4$ points are successively set , $A, P, Q,W $, which are the points of intersection of the bisector $AL $ of the triangle $ABC$ with the circumscribed and inscribed circle. Knowing only these points, construct a triangle $ABC $.

Consider a completely inelastic collision between two lumps of space goo. Lump 1 has mass $m$ and originally moves directly north with a speed $v_\text{0}$. Lump 2 has mass $3m$ and originally moves directly east with speed $v_\text{0}/2$. What is the final speed of the masses after the collision? Ignore gravity, and assume the two lumps stick together after the collision. (A) $7/16 \ v_\text{0}$ (B) $\sqrt{5}/8 \ v_\text{0}$ (C) $\sqrt{13}/8 \ v_\text{0}$ (D) $5/8 \ v_\text{0}$ (E) $\sqrt{13/8} \ v_\text{0}$

Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.

Let $k, n$ be odd positve integers greater than $1$. Prove that if there a exists natural number $a$ such that $k|2^a+1, \ n|2^a-1$, then there is no natural number $b$ satisfying $k|2^b-1, \ n|2^b+1$.

Let $ABCD$ be a square of side length $13$. Let $E$ and $F$ be points on rays $AB$ and $AD$ respectively, so that the area of square $ABCD$ equals the area of triangle $AEF$. If $EF$ intersects $BC$ at $X$ and $BX = 6$, determine $DF$.

What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$?

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

Let $ABCD$ be a rectangle. Construct equilateral triangles $BCX$ and $DCY$, in such a way that both of these triangles share some of their interior points with some interior points of the rectangle. Line $AX$ intersects line $CD$ on $P$, and line $AY$ intersects line $BC$ on $Q$. Prove that triangle $APQ$ is equilateral.

Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation $$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$ Calculate $ \int_{-2019}^{2019}f(x)dx . $ [i]Constantin Rusu[/i]

How many quadrilaterals in the plane have four of the nine points $(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)$ as vertices? Do count both concave and convex quadrilaterals, but do not count figures where two sides cross each other or where a vertex angle is $180^{\circ}$. Rigorously verify that no quadrilateral was skipped or counted more than once. [asy]size(50); dot((0,0)); dot((0,1)); dot((0,2)); dot((1,0)); dot((1,1)); dot((1,2)); dot((2,0)); dot((2,1)); dot((2,2));[/asy]

Let $ \vartriangle ABC $ be a triangle with $ \angle ACB = 60º $. Let $ E $ be a point inside $ \overline {AC} $ such that $ CE <BC $. Let $ D $ over $ \overline {BC} $ such that $$ \frac {AE} {BD} = \frac {BC} {CE} -1 .$$ Let us call $ P $ the intersection of $ \overline {AD} $ with $ \overline {BE} $ and $ Q $ the other point of intersection of the circumcircles of the triangles $ AEP $ and $ BDP $. Prove that $QE \parallel BC $.

What triangles can be cut into three triangles having equal radii of circumcircles?

Let a polynomial $p(x)$ with integer coefficients take the value $5$ for five different integer values of $x.$ Prove that $p(x)$ does not take the value $8$ for any integer $x.$

Find a sequence of positive integers $f(n)$ ($n \in \mathbb{N}$) such that: (i) $f(n) \leq n^8$ for any $n \geq 2$; (ii) for any distinct $a_1, \cdots, a_k, n$, $f(n) \neq f(a_1) + \cdots+ f(a_k)$.