Let $ A^{\prime}$ be an arbitrary point on the side $ BC$ of a triangle $ ABC$. Denote by $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ the circles simultanously tangent to $ AA^{\prime}$, $ A^{\prime}B$, $ \Gamma$ and $ AA^{\prime}$, $ A^{\prime}C$, $ \Gamma$, respectively, where $ \Gamma$ is the circumcircle of $ ABC$. Prove that $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ are congruent if and only if $ AA^{\prime}$ passes through the Nagel point of triangle $ ABC$. ([i]If $ M,N,P$ are the points of tangency of the excircles of the triangle $ ABC$ with the sides of the triangle $ BC$, $ CA$ and $ AB$ respectively, then the Nagel point of the triangle is the intersection point of the lines $ AM$, $ BN$ and $ CP$[/i].)

Prove the following inequality: $tan \, 1>\frac{3}{2}$.

Using only prime numbers, a set is formed with the following conditions: Any one-digit prime number can be in the set. For a prime number with more than one digit to be in the set, the number that results from deleting only the first digit and also the number that results from deleting only the last digit must be in the set. Write, of the sets that meet these conditions, the one with the greatest number of elements. Justify why there cannot be one with more elements. Remember that the number $1$ is not prime.

At some integer points a polynomial with integer coefficients take values $1, 2$ and $3$. Prove that there exist not more than one integer at which the polynomial is equal to $5$.

A point $ P$ is performing a random walk on the $ X$-axis. At the instant $ t\equal{}0$, $ P$ is at a point $ x_0$ ($ |x_0|\le N$, where $ x_0$ and $ N$ denote integers, $ N>0$). If at an instant $ t$ ($ t$ being a nonnegative integer), $ P$ is at a point of $ x$ integer abscissa and $ |x|<N$, then by the instant $ t\plus{}1$ it reaches either the point $ x\plus{}1$ or the point $ x\minus{}1$, each with probability $ \frac12$. If at the instant $ t$, $ P$ is at the point $ x\equal{}N$ [$ x\equal{}\minus{}N$], then by the instant $ t\plus{}1$ it is certain to reach the point $ N\minus{}1$ [$ \minus{}N\plus{}1$]. Denote by $ P_k(t)$ the probability of $ P$ being at $ x\equal{}k$ at instant $ t$ ($ k$ is an integer). Find $ \lim_{t\to \infty}P_{k}(2t)$ and $ \lim_{t\to \infty}P_k(2t\plus{}1)$ for every fixed $ k$.

Circles $C_1$ and $C_2$ intersect at exactly two points $I_1$ and $I_2$. A point $J$ on $C_1$ outside of $C_2$ is chosen such that $\overline{JI_2}$ is tangent to $C_2$ and $\overline{JI_2} = 3$. A line segment is drawn from $J$ through $I_1$ and intersects $C_2$ at point $K$ and $\overline{JK} = 6$. $\angle JI_2I_1 = \angle I_2KI_1 = \frac12 \angle I_1I_2K$. Let $\overline{I_1I_2} = a$, and let $a$ equal the fraction$ \frac{m\sqrt{p}}{n}$ , where $m$ and $n$ are coprime and $p$ is a positive integer not divisible by the square of any prime. Find $100m + 10p + n$.

Michelle is birdwatching. At time $t=0$, she spots $n$ birds all standing on a power cable, in a single line. Every minute after she first spots the birds, she looks back up at the birds, counting the number of them that are left. Assume that each minute, each bird has a $50\%$ chance to fly off, and that no birds decide to perch on the cable for $t\geq0$. As $n$ approaches $\infty$, let the probability that Michelle will see exactly $1$ bird on the line at some point in time approach $p$. Estimate $\lfloor 10000p \rfloor$. \\\\ Your score will be calculated by the function $\max(0, \lfloor20 - \frac{|A - S|}{12}\rfloor)$, where $S$ is your submission and $A$ is the true answer. [i]Lightning 5.1[/i]

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

Without using tables, find the exact value of the product: \[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]

Solve the recurrence $R_0=1, R_n=nR_{n-1}+2^n\cdot n!$.

Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions: [b]a)[/b] $ f''>0\implies f-\alpha >0 $ [b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $

Suppose the transformation $T$ acts on points in the plane like this: $$T(x, y) = \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2 + y^2}\right).$$ Determine the area enclosed by the set of points of the form $T(x, y)$, where $(x, y)$ is a point on the edge of a length-$2$ square centered at the origin with sides parallel to the axes.

Let $S$ be a sequence $n_1, n_2,..., n_{1995}$ of positive integers such that $n_1 +...+ n_{1995 }=m < 3990$. Prove that for each integer $q$ with $1 \le q \le m$, there is a sequence $n_{i_1} , n_{i_2} , ... , n_{i_k}$ , where $1 \le i_1 < i_2 < ...< i_k \le 1995$, $n_{i_1} + ...+ n_{i_k} = q$ and $k$ depends on $q$.

A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours

On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.

A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.

The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. The total surface area of the box is $\text{(A)}\ 776 \qquad \text{(B)}\ 784 \qquad \text{(C)}\ 798 \qquad \text{(D)}\ 800 \qquad \text{(E)}\ 812$

Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$, $AO = CO$. Points $P$ and $Q$ are marked on the segments $AO$ and $CO$, respectively, such that $PO = OQ$. Let $N$ and $K$ be the intersection points of the sides $AB$, $CD$, and the lines $DP$ and $BQ$ respectively. Prove that the points $N$, $O$, and $K$ are colinear.

We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?

Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and $$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$ Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$

$ N $ grasshoppers are lined up in a row. At any time, one grasshopper is allowed to jump over exactly two grasshoppers standing to the right or left of him. At what $ n $ can grasshoppers rearrange themselves in reverse order?

$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root. $(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.

Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages $ 40$ miles per hour, he arrives at his workplace three minutes late. When he averages $ 60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 58$

Weights of $1$ g, $2$ g,$ ...$ , $200$ g are placed on the two pans of a balance such that on each pan there are $100$ weights and the balance is in equilibrium. Prove that one can swap $50$ weights from one pan with $50$ weights from the other pan such that the balance remains in equilibrium. Kvant Magazine

In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2$ . (here $r$ is the radius of the circle inscribed in the triangle $ABC$). (Mykola Moroz)