In the quadrilateral $ABCD$, we have $\measuredangle BAD = 100^{\circ}$, $\measuredangle BCD = 130^{\circ}$, and $AB=AD=1$ centimeter. Find the length of diagonal $AC$.

For all positive reals $a$ and $b$, show that $$\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab > 18.$$

Find all positive integers $n$ such that the number $$n^6 + 5n^3 + 4n + 116$$is the product of two or more consecutive numbers.

Each cell of an infinite table (infinite in all directions) is colored with one of $n$ given colors. All six cells of any $2\times 3$ (or $3 \times 2$) rectangle have different colors. Find the smallest possible value of $n$.

For real numbers $a$ and $b$, define $$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$

Let $ABC$ be a triangle and $AA_1$ be the angle bisector of $A$ ($A_1 \in BC$). The point $P$ is on the segment $AA_1$ and $M$ is the midpoint of the side $BC$. The point $Q$ is on the line connecting $P$ and $M$ such that $M$ is the midpoint of $PQ$. Define $D$ and $E$ as the intersections of $BQ$, $AC$, and $CQ$, $AB$. Prove that $CD=BE$.

Prove for all positive real numbers $m,n,p,q$ that $$\frac{m}{t+n+p+q} + \frac{n}{t+p+q+m} + \frac{p}{t+q+m+n} + \frac{q}{t+m+n+p} \geq \frac{4}{5},$$where $t=\frac{m+n+p+q}{2}.$

The real numbers $a_1 \leq a_2 \leq \cdots \leq a_{672}$ are given such that $$a_1 + a_2 + \cdots + a_{672} = 2018.$$For any $n \leq 672$, there are $n$ of these numbers with an integer sum. What is the smallest possible value of $a_{672}$?

Taken from Pregătire Matematică Olimpiade Juniori. Translated by Amir Hossein Parvardi.