Exploring Absolute Values and Inequalities in Geometry and Algebra

Solve the equation

$|2026 - x| + |x - 2026| = 2025.$

1. Solve the inequality

   $|x + 2025| < |x - 2026|.$

2. Prove that if $a + b + c + d > 0$, $a > c$, and $b > d$, then

   $|a + b| > |c + d|.$

3. Prove that there are no real numbers $x, y, t$ such that all three inequalities hold simultaneously:

   $|x| < |y - t|, \quad |y| < |t - x|, \quad |t| < |x - y|.$

4. On the coordinate plane, plot all points whose coordinates satisfy the equation

   $y^2 - |y| = x^2 - |x|.$

5. Sophie asserts that for any three points $A, B,$ and $C$ in the plane, the inequality

   $AB \geq |AC - BC|$

   holds. Prove that she is correct.

6. Emma claims that in any triangle, the length of any side does not exceed the semiperimeter. Prove that she is right.

7. Two rivers (straight lines)—one with fresh water and one with salt water—intersect at an acute angle. Inside this acute angle stands Tom, who wishes to collect both fresh and salt water and return to his starting point. How should he do this while travelling the minimum possible distance?