2023 SSMO Accuracy Round Problems

Problem 1

Mr. Sammy proposes a Hamburger Proclamation, which has lines, divided into paragraphs of lines each. It takes him seconds to read each line. Additionally, he adds a second pause between two lines in a paragraph, and a second pause between paragraphs. If it takes him minutes to read the whole Hamburger Proclamation, find

Solution

Problem 2

Suppose that the average of all -digit palindromes is denoted by and the average of all -digit numbers is denoted by Find

Solution

Problem 3

Suppose that are real numbers such \begin{align*}

   a + b - c &= 4 \\
   a^2 + b^2 + c^2 &= 14 \\
   a^3 + b^3 - c^3 &= 34 \\

\end{align*} Find the sum of all possible values of .

Solution

Problem 4

In square point is selected on diagonal Let be the intersection of the circumcircles of triangles and Given that and find the maximum possible area of triangle (A circumcircle of some triangle is the circle containing , , and )

Solution

Problem 5

Define the between two numbers and to be where is the number of divisors of . Find the sum of integers which have a relationship of with .

Solution

Problem 6

Let the roots of be . Find \[

   \frac{p^2 + q^2}{p + q} + \frac{q^2 + r^2}{q + r} + \frac{r^2 + p^2}{r + p}

\]

Solution

Problem 7

Concentric circles and are drawn, with radii and respectively. Chords and of are both tangent to and intersect at If then the sum of all possible distinct values of can be expressed as for relatively prime positive integers and Find

Solution

Problem 8

There is a quadrilateral inscribed in a circle with center . In quadrilateral , diagonal is a diameter of the circle, and Let be the base of the altitude from onto side . Let be the base of the altitude from onto . Given that and that the product of the lengths of the diagonals of is for some squarefree find

Solution

Problem 9

Consider a grid called . We take one of the four smaller grids located in as . We repeat the process of taking smaller grids until we eventually converge at the unit square

\begin{center}

   \begin{asy}
       size(7cm);
       filldraw((0,0)--(0,10)--(10,10)--(10,0)--cycle, opacity(0.2)+lightblue, blue);
       fill((0,0)--(0,9)--(9,9)--(9,0)--cycle, 
       opacity(0.1)+lightblue);
       draw((0,9)--(9,9)--(9,0), 
       blue);
       fill((1,0)--(1,8)--(9,8)--(9,0)--cycle, opacity(0.1)+lightblue);
       draw((1,0)--(1,8)--(9,8), blue);

       label("", (8.3, 9.52));
       label("", (6.8, 8.52));
       label("", (4.78, 7.52));

   \end{asy}

\end{center}

Of the distinct tuples of shrinking grids , let be the number of these tuples such that their last element is the center square of the original grid . Find the largest integer such Solution

Problem 10

Let be a triangle such , , . Let the incircle of touch at , at , and at . Let be the line through the midpoints of and . Define and similarily. Let the area of the star created by the union of and the triangle bound by , , and be for relatively prime and . Find .

Solution