2025 Academic Year College Scholastic Ability Test – Geometry

What is the value of ?

For the function , what is the value of ?

For a geometric sequence where both the first term and common ratio are positive , if it satisfies what is the value of ?

When the function is continuous on the set of all real numbers, what is the value of the constant ?

For the function , what is the value of ?

When , what is the value of ?

When a polynomial function satisfies for all real numbers , what is the value of ?

For two real numbers , , what is the value of ?

For the function , when , what is the value of the positive number ?

For the function defined on the closed interval , find the minimum value of for ordered pairs of natural numbers such that the function has a maximum value of at .

A point P moves along a number line starting at time . The position of point P at time is given by What is the acceleration of point P at the time when the direction of motion changes after starting?

Given a sequence with and an arithmetic sequence with that satisfy for all natural numbers , find the value of .

A cubic function with leading coefficient satisfies For the origin and point , let be the point where line segment intersects the curve , other than point . Let be the area enclosed by the curve , the -axis, and line segment , and let be the area enclosed by the curve and line segment . Find the value of .

As shown in the figure, in triangle ABC, point D is taken on segment AB such that , and let O be the circle centered at point A and passing through point D, and let E be the point where circle O intersects segment AC. Given that , and the ratio of the areas of triangle ADE and triangle ABC is . When the radius of the circumcircle of triangle ABC is , what is the maximum value of the area of triangle PBC for a point P on circle O? (Note: )

For a constant and a quadratic function with a negative leading coefficient, the function satisfies the following conditions. (a) The function is differentiable on the entire set of real numbers. (b) The number of distinct real roots of the equation with respect to is . What is the value of ?

Find the value of the real number that satisfies the equation

Given that the sequence satisfies for all natural numbers , find the value of .

Let be the -coordinate of the intersection point of the curve and the line . A function defined on the set of all real numbers satisfies the following conditions. For all real numbers such that , and . Find the value of .

For all sequences where all terms are integers and satisfy the following conditions, find the sum of all possible values of . (a) For all natural numbers , (b) The minimum value of the natural number such that is .

For two vectors , when , what is the value of ?

When a parabola with vertex at and directrix passes through the point , what is the value of the positive number ?

For two points A(a, b, 6) and B(-4, -2, c) in coordinate space, when the point that internally divides segment AB in the ratio 3:2 lies on the z-axis, and the point that externally divides segment AB in the ratio 3:2 lies on the xy-plane, what is the value of a + b + c?

For a natural number , let the line intersect the two ellipses at points and respectively in the first quadrant. Let be the -intercept of the tangent line to ellipse at point , and be the -intercept of the tangent line to ellipse at point . Find the number of all values of such that .

As shown in the figure, for tetrahedron ABCD with and , let M be the midpoint of segment BC. When triangle AMD is an equilateral triangle and line BC is perpendicular to plane AMD, what is the area of the orthogonal projection of the circle inscribed in triangle ACD onto plane BCD?

In coordinate space, there is a right triangle ABC with , , , and a sphere with diameter . A plane that contains line and is perpendicular to plane intersects sphere to form a circle . Among the points on circle , let and be two distinct points whose distance to line is . What is the length of segment ?

In the coordinate plane, there is a square ABCD with side length 4. Let S be the figure formed by points X that satisfy For a point P on figure S, let Q be the point that satisfies Let M and m be the maximum and minimum values of , respectively. Find the value of .

There is a hyperbola with two foci and where . For a point on this hyperbola in the first quadrant, take a point on line such that . When triangles and are similar to each other, the area of triangle is . Find the value of . (Given that and and are coprime natural numbers.)