Let $ABCDEF$ be a convex hexagon satisfying $AC=DF$, $CE=FB$ and $EA=BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

In a triangle $\triangle ABC$ with $\angle ABC < \angle BCA$, we define $K$ as the excenter with respect to $A$. The lines $AK$ and $BC$ intersect in a point $D$. Let $E$ be the circumcenter of $\triangle BKC$. Prove that \[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]

For a fixed triangle $ABC$ we choose a point $M$ on the ray $CA$ (after $A$), a point $N$ on the ray $AB$ (after $B$) and a point $P$ on the ray $BC$ (after $C$) in a way such that $AM -BC = BN- AC = CP – AB$. Prove that the angles of triangle $MNP$ do not depend on the choice of $M, N, P$ .

There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find the least possible value of $b + c.$

Three points of a triangle are among 8 vertex of a cube. So the number of such acute triangles is $\text{(A)}0\qquad\text{(B)}6\qquad\text{(C)}8\qquad\text{(D)}24$

Determine the integers $k\geq 2$ for which the sequence $\Big\{ \binom{2n}{n} \pmod{k}\Big\}_{n\in \mathbb{Z}_{\geq 0}}$ is eventually periodic.

Given positive integer $M.$ For any $n\in\mathbb N_+,$ let $h(n)$ be the number of elements in $[n]$ that are coprime to $M.$ Define $\beta :=\frac {h(M)}M.$ Proof: there are at least $\frac M3$ elements $n$ in $[M],$ satisfy $$\left| h(n)-\beta n\right|\le\sqrt{\beta\cdot 2^{\omega(M)-3}}+1.$$ Here $[n]:=\{1,2,\ldots ,n\}$ for all positive integer $n.$ [i]Proposed by Bin Wang[/i]

Let $ S$ be the set of $25$ points $(x, y)$ with $0\le x, y \le 4$. A triangle whose three vertices are in $S$ is chosen at random. What is the expected value of the square of its area?

The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

Juan found an easy (but wrong) way to simplify fractions. He proposes to simplify a fraction $\frac{M}{N}$ , where $M <N$ are two natural numbers, erase simultaneously the equal digits in the numerator and denominator. For instance, $\frac{12356}{5789}$ transforms after simplification of Juan in $\frac{126}{789}$. Find out if there is at least one fraction $\frac{M}{N}$, with $10 <M <N <100$ for which this method gives a correct result.

The quadrilateral $A_1A_2A_3A_4$ is cyclic, and its sides are $a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4$ and $a_4 = A_4A_1.$ The respective circles with centres $I_i$ and radii $r_i$ are tangent externally to each side $a_i$ and to the sides $a_{i+1}$ and $a_{i-1}$ extended. ($a_0 = a_4$). Show that \[ \prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2. \]

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.

Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent: [list=1] [*]There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;[/*] [*]$\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.[/*] [/list] [i]Proposed by Tomáš Bárta, Charles University, Prague[/i]

Three equal mass satellites $A$, $B$, and $C$ are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are $L_A$, $L_B$, and $L_C$. Which of the following statements is correct? [asy] // Code created by riben size(250); dotfactor=12; draw(circle((0,0),1.5),linewidth(2)); draw(circle((0,0),6),dashdotted); draw(circle((0,0),14),dashed); draw(ellipse((4,0),10,8),linewidth(1)); pair A,B,C; A=(-7,12.12); B=(5,7.9); C=(5.7,-1.87); dot(A); dot(B); dot(C); label("A",A,NW*1.5); label("B",B,NW*1.5); label("C",C,E*1.5); filldraw((-1.500, 0.078)-- (-1.428, 0.080)-- (-1.337, 0.094)-- (-1.295, 0.157)-- (-1.246, 0.209)-- (-1.186, 0.227)-- (-1.143, 0.290)-- (-1.148, 0.357)-- (-1.135, 0.469)-- (-1.057, 0.505)-- (-0.996, 0.563)-- (-0.936, 0.526)-- (-0.852, 0.557)-- (-0.773, 0.587)-- (-0.772, 0.716)-- (-0.765, 0.828)-- (-0.781, 0.955)-- (-0.732, 1.035)-- (-0.648, 1.083)-- (-0.605, 1.162)-- (-0.604, 1.246)-- (-0.645, 1.295)-- (-0.736, 1.270)-- (-0.796, 1.229)-- (-0.851, 1.193)-- (-0.941, 1.135)-- (-1.014, 1.076)-- (-1.105, 0.995)-- (-1.154, 0.921)-- (-1.227, 0.841)-- (-1.288, 0.760)-- (-1.349, 0.669)-- (-1.398, 0.556)-- (-1.453, 0.465)-- (-1.485, 0.357)-- (-1.510, 0.239)--cycle,gray); filldraw((-0.119, 1.245)-- (-0.130, 1.193)-- (-0.146, 1.095)-- (-0.202, 1.056)-- (-0.327, 1.033)-- (-0.262, 1.031)-- (-0.278, 0.979)-- (-0.193, 0.949)-- (-0.108, 0.943)-- (-0.013, 0.941)-- (0.032, 0.915)-- (0.026, 0.840)-- (0.015, 0.779)-- (0.019, 0.705)-- (0.074, 0.646)-- (0.113, 0.582)-- (0.162, 0.533)-- (0.167, 0.463)-- (0.241, 0.400)-- (0.311, 0.412)-- (0.416, 0.410)-- (0.465, 0.342)-- (0.541, 0.410)-- (0.611, 0.347)-- (0.679, 0.242)-- (0.728, 0.132)-- (0.732, 0.048)-- (0.671, -0.037)-- (0.615, -0.104)-- (0.540, -0.172)-- (0.409, -0.209)-- (0.324, -0.244)-- (0.253, -0.293)-- (0.188, -0.314)-- (0.162, -0.389)-- (0.181, -0.486)-- (0.270, -0.534)-- (0.340, -0.537)-- (0.380, -0.596)-- (0.424, -0.688)-- (0.418, -0.772)-- (0.352, -0.825)-- (0.281, -0.883)-- (0.241, -0.926)-- (0.145, -0.981)-- (0.044, -1.044)-- (-0.006, -1.107)-- (-0.007, -1.190)-- (0.077, -1.216)-- (0.162, -1.213)-- (0.253, -1.163)-- (0.323, -1.128)-- (0.404, -1.075)-- (0.510, -1.015)-- (0.605, -0.980)-- (0.671, -0.931)-- (0.731, -0.920)-- (0.817, -0.852)-- (0.898, -0.798)-- (0.963, -0.777)-- (0.964, -0.708)-- (1.024, -0.645)-- (1.025, -0.571)-- (0.976, -0.488)-- (0.912, -0.425)-- (0.878, -0.347)-- (0.823, -0.289)-- (0.779, -0.225)-- (0.744, -0.193)-- (0.756, -0.100)-- (0.816, -0.033)-- (0.837, 0.047)-- (0.838, 0.122)-- (0.824, 0.200)-- (0.800, 0.307)-- (0.796, 0.381)-- (0.872, 0.416)-- (0.967, 0.414)-- (1.016, 0.360)-- (1.096, 0.381)-- (1.117, 0.428)-- (1.058, 0.506)-- (0.998, 0.564)-- (0.954, 0.591)-- (0.914, 0.617)-- (0.860, 0.676)-- (0.800, 0.716)-- (0.751, 0.775)-- (0.757, 0.859)-- (0.797, 0.921)-- (0.823, 0.987)-- (0.889, 1.096)-- (0.850, 1.160)-- (0.780, 1.176)-- (0.700, 1.183)-- (0.645, 1.125)-- (0.579, 1.039)-- (0.518, 0.986)-- (0.438, 0.956)-- (0.343, 0.967)-- (0.289, 1.049)-- (0.249, 1.117)-- (0.195, 1.176)-- (0.125, 1.192)-- (0.030, 1.208)-- (-0.040, 1.220)--cycle,gray); [/asy] (A) $L_\text{A} > L_\text{B} > L_\text{C}$ (B) $L_\text{C} > L_\text{B} > L_\text{A}$ (C) $L_\text{B} > L_\text{C} > L_\text{A}$ (D) $L_\text{B} > L_\text{A} > L_\text{C}$ (E) The relationship between the magnitudes is different at various instants in time.

$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.

How many integers of the form $n^{2023-n}$ are perfect squares, where $n$ is a positive integer between $1$ and $2023$ inclusive?