In the triangle $ ABC $, the bisectors of the internal and external angles are drawn at the vertices $ A $ and $ B $. Prove that the orthogonal projections of the point $ C $ on these bisectors lie on one straight line.

Determine all functions $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for any positive reals $x,y$, $$f(xy+f(xy)) = xf(y) + yf(x)$$

Let $P$ be a point in the interior of the triangle $ABC$. The lines $AP$, $BP$, and $CP$ intersect the sides $BC$, $CA$, and $AB$ at $D,E$, and $F$, respectively. A point $Q$ is taken on the ray $[BE$ such that $E\in [BQ]$ and $m(\widehat{EDQ})=m(\widehat{BDF})$. If $BE$ and $AD$ are perpendicular, and $|DQ|=2|BD|$, prove that $m(\widehat{FDE})=60^\circ$.

(1) Evaluate $ \int_1^{3\sqrt{3}} \left(\frac{1}{\sqrt[3]{x^2}}\minus{}\frac{1}{1\plus{}\sqrt[3]{x^2}}\right)\ dx.$ (2) Find the positive real numbers $ a,\ b$ such that for $ t>1,$ $ \lim_{t\rightarrow \infty} \left(\int_1^t \frac{1}{1\plus{}\sqrt[3]{x^2}}\ dx\minus{}at^b\right)$ converges.

Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that: [LIST] [*] in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and [/*] [*]in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.[/*] [/LIST] [b]Note.[/b] The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.

Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find \[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]

How many even integers are there between 200 and 700 whose digits are all different and come from the set {1,2,5,7,8,9}? $\textbf{(A)}\,12 \qquad\textbf{(B)}\,20 \qquad\textbf{(C)}\,72 \qquad\textbf{(D)}\,120 \qquad\textbf{(E)}\,200$

Starting at $12:00:00$ AM on January $1,$ $2022,$ after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365.$ Find $y + d.$

A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true? (B. Frenkin)

Let $a_n$ be a sequence of reals. Suppose $\sum a_n$ converges. Do these sums converge aswell? (a) $a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+...$ (b) ${a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}$

Two cyclists start moving simultaneously in opposite directions on a circular circuit. The first cyclist maintains a constant speed $c_1$ meters per second, the second maintains $c_2$ meters per second. How many times did they meet when the first cyclist completed $n$ laps? Compute for $c_1=10,c_2=7,n=11$.

Find all monotonic functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[f(f(x) - y) + f(x+y) = 0,\] for every real $x, y$. (Note that monotonic means that function is not increasing or not decreasing)

Can the touchpoints of the inscribed circle of a triangle with the triangle form an obtuse triangle?

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$? [i]Proposed by Oleksiy Masalitin[/i]

On the side $AC$ of triangle $ABC$ point $D$ is chosen. Let $I_1, I_2, I$ be the incenters of triangles $ABD, BCD, ABC$ respectively. It turned out that $I$ is the orthocentre of triangle $I_1I_2B$. Prove that $BD$ is an altitude of triangle $ABC$.

One country consists of islands $A_1,A_2,\cdots,A_N$,The ministry of transport decided to build some bridges such that anyone will can travel by car from any of the islands $A_1,A_2,\cdots,A_N$ to any another island by one or more of these bridges. For technical reasons the only bridges that can be built is between $A_i$ and $A_{i+1}$ where $i = 1,2,\cdots,N-1$ , and between $A_i$ and $A_N$ where $i<N$. We say that a plan to build some bridges is good if it is satisfies the above conditions , but when we remove any bridge it will not satisfy this conditions. We assume that there is $a_N$ of good plans. Observe that $a_1 = 1$ (The only good plan is to not build any bridge) , and $a_2 = 1$ (We build one bridge). 1-Prove that $a_3 = 3$ 2-Draw at least $5$ different good plans in the case that $N=4$ and the islands are the vertices of a square 3-Compute $a_4$ 4-Compute $a_6$ 5-Prove that there is a positive integer $i$ such that $1438$ divides $a_i$

Let $P_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \ge 0$ by $$x_{n+1} = - \frac{3x_n - y_n}{2}, y_{n+1} = - \frac{x_n + y_n}{2}$$Find the area of the quadrilateral formed by the points $P_{96}, P_{97}, P_{98}, P_{99}$.

Find all positive integers N with at most 4 digits such that the number obtained by reversing the order of digits of N is divisible by N and differs from N.

In $\triangle ABC$, $AB>AC$. Let $O$ and $I$ be the circumcenter and incenter respectively. Prove that if $\angle AIO = 30^{\circ}$, then $\angle ABC = 60^{\circ}$.

Twelve couples are seated around a circular table such that all of men are seated side by side, and every women are seated to opposite of her husband. In every step, a woman and a man next to her are swapping. What is the least possible number of swapping until all couples are seated side by side? $\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 55 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 66 \qquad\textbf{(E)}\ \text{None}$

Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ : $$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$

If 1 pint of paint is needed to paint a statue 6 ft. high, then the number of pints it will take to paint (to the same thickness) 540 statues similar to the original but only 1 ft. high is $ \textbf{(A)}\ 90 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 15$

Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. What is the probability that Flora will eventually land at $10$? $\textbf{(A) } \frac{5}{512} \qquad \textbf{(B) } \frac{45}{1024} \qquad \textbf{(C) } \frac{127}{1024} \qquad \textbf{(D) } \frac{511}{1024} \qquad \textbf{(E) } \frac{1}{2}$

Regular hexagon $ABCDEF$ has side length $2$. Line segment $BD$ is drawn, and circle $O$ is inscribed inside the pentagon $ABDEF$ such that $O$ is tangent to $AF$, $BD$, and $EF$. Compute the radius of $O$.