2022 AMC 10 A Problem 25

Lattice-Point Counting and Integer-Equation Method

Count lattice points by rows and columns, including the boundary. The ratio for $R$ and $S$ gives a common integer parameter, while the intersection ratio fixes the horizontal placement of $T$.

Step 1: Parameterize the sizes of the two large squares

Let the edge lengths of $R$ and $S$ be positive integers $r$ and $s$. Each has one more lattice row and one more lattice column than its edge length. Because the point-count ratio is the square of a positive rational ratio, the corresponding numbers of rows are in the ratio $3:2$.

Step 2: Count the lattice points in the union

The squares share the part of the $y$-axis from height zero through the top of $S$. Since $S$ is shorter, this common boundary contains one point for every integer height from zero through $s$. Inclusion–exclusion then gives the union count.

Step 3: Show that the top of T cannot lie above S

Let $t$ be the edge length of $T$. If its top were above the top of $S$, both top vertices would have to lie in $R$, because at that height the union has no points to the left of the $y$-axis. Then the overlap with $S$ could only occur along the $y$-axis, while the overlap with $R$ would contain at least as many points. This contradicts the condition derived below that the overlap with $S$ is twelve times as large. Therefore the top of $T$ is no higher than the top of $S$, so every point of $T$ lies in the continuous region $R\cup S$.

Step 4: Translate the intersection-fraction condition

Let $A$ and $B$ be the numbers of lattice points in the intersections with $R$ and $S$, respectively. Substitute the known total point counts. The stated factor of $27$ then simplifies to a direct ratio between the two intersection counts.

Step 5: Count the shared axis inside T

Both intersection counts are positive: otherwise their ratio condition and the placement of the top vertices would be impossible. Thus $T$ reaches both sides of the $y$-axis and contains that axis. Its common portion with both $R$ and $S$ consists of one vertical lattice column containing one point at each integer height. Since $T$ lies in the union, inclusion–exclusion counts all points of $T$.

Step 6: Determine the horizontal placement of T

Write the horizontal interval of $T$ as running from the integer coordinate $a$ to the integer coordinate $q$, where $a\le 0\le q$. In $R$, there are $q+1$ lattice columns; in $S$, there are $-a+1$ lattice columns. Every such column has the same number of lattice points because $T$ has full height in both larger squares.

Step 7: Introduce the placement parameter

Let the number of lattice columns of $T$ in $R$ be $v$. Dividing the formula for $A$ by the positive number of points in each column relates $v$ to the height. The left-side column ratio then determines both endpoints and the edge length of $T$.

Step 8: Use the one-fourth condition

The point count of $T$ is one fourth of the union count. Substituting the two integer parameters produces the equation that all legal configurations must satisfy.

Step 9: Find the first possible value of the placement parameter

Treat the equation as a quadratic in $k$. Its discriminant must be a perfect square. For each smaller positive value of $v$, the resulting number lies strictly between the consecutive squares displayed, so none can produce an integer $k$. The first successful value is $v=7$.

Step 10: Prove that this solution gives the global minimum

The preceding check eliminates every smaller positive value of $v$. Any value at least eight already forces a sum larger than the candidate, even if only the weak lower bound on $k$ is used. Thus no later solution of the integer equation can improve the result.

Step 11: Construct the attaining configuration

Use the successful parameters to obtain all three edge lengths. Place $R$ to the right of the axis and $S$ to the left as required. Place $T$ with the calculated horizontal endpoints; its top is below the top of $S$, so both top vertices are in the union. This gives a legal configuration attaining the bound.