Let \( x \) be a real number. Determine the solution to the following equation: \[ \frac{x^2 + 1}{1} + \frac{x^2 + 2}{2} + \frac{x^2 + 3}{3} + \ldots + \frac{x^2 + 2024}{2024} = 2024 \]

There are 50 slips of paper numbered from 1 to 50. It is desired to pick 3 slips such that for any of the three numbers, divided by the greatest common divisor of the other two, the square root of the result is a rational number. How many unordered triples of slips satisfy this condition?

Let \( ABC \) be a triangle and \( D \) the foot of the altitude from \( A \). Let \( M \) be a point such that \( MB = MC \). Let \( E \) and \( F \) be the intersections of the circumcircle of \( BMD \) and \( CMD \) with \( AD \). Let \( G \) and \( H \) be the intersections of \( MB \) and \( MC \) with \( AD \). Prove that \( EG = FH \).

There are 6 squares in a row. Each one is labeled with the name of Ana or Beto and with a number from 1 to 6, using each number without repetition. Ana and Beto take turns painting each square according to the order of the numbers on the labels. Whoever paints the square will be the person whose name is on the label. When painting, the person can choose to paint the square either red or blue. Beto wins if at the end there are the same number of blue squares as red squares, and Ana wins otherwise. In how many of all the possible ways of labeling the squares can Beto ensure his victory? The following is an example of a labeling of the labels. [asy][asy] size(12cm); draw((0,0)--(6,0)--(6,-1)--(0,-1)--cycle); for (int i=1; i<6; ++i) { draw((i,0)--(i,-1)); } for (int i=1; i<6; ++i) { draw((i,0)--(i,-1.25)); } draw((0,0)--(6,0)--(6,-1.25)--(0,-1.25)--cycle); for (int i=1; i<7; ++i) { draw((i-0.5,-1)--(i-0.5,-1.25)); } label("Ana", (0.25, -1.125)); label("Beto", (1.25, -1.125)); label("Ana", (2.25, -1.125)); label("Beto", (3.25, -1.125)); label("Ana", (4.25, -1.125)); label("Beto", (5.25, -1.125)); label("1", (0.75, -1.125)); label("3", (1.75, -1.125)); label("5", (2.75, -1.125)); label("2", (3.75, -1.125)); label("4", (4.75, -1.125)); label("6", (5.75, -1.125)); [/asy][/asy] First Ana paints the first square, then Beto paints the fourth square, then Beto paints the second square, and so on.

Consider the acute-angled triangle \(ABC\). The segment \(BC\) measures 40 units. Let \(H\) be the orthocenter of triangle \(ABC\) and \(O\) its circumcenter. Let \(D\) be the foot of the altitude from \(A\) and \(E\) the foot of the altitude from \(B\). Additionally, point \(D\) divides the segment \(BC\) such that \(\frac{BD}{DC} = \frac{3}{5}\). If the perpendicular bisector of segment \(AC\) passes through point \(D\), calculate the area of quadrilateral \(DHEO\).

On a \(4 \times 4\) board, each cell is colored either black or white such that each row and each column have an even number of black cells. How many ways can the board be colored?

Consider the quadratic equation \(x^2 + a_0 x + b_0\) for some real numbers \((a_0, b_0)\). Repeat the following procedure as many times as possible: Let \(c_i = \min \{r_i, s_i\}\), with \(r_i, s_i\) being the roots of the equation \(x^2 + a_i x + b_i\). The new equation is written as \(x^2 + b_i x + c_i\). That is, for the next iteration of the procedure, \(a_{i+1} = b_i\) and \(b_{i+1} = c_i\). We say that \((a_0, b_0)\) is an $\textit{interesting}$ pair if, after a finite number of steps, the equation we obtain after one step is the same, so that \((a_i, b_i) = (a_{i+1}, b_{i+1})\). Find all $\textit{interesting}$ pairs.

Find all positive integers \(n\) such that among the \(n\) numbers \[ 2n + 1, \, 2^2 n + 1, \, \ldots, \, 2^n n + 1 \] there are \(n\), \(n - 1\), or \(n - 2\) primes.