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Found: 3

Problems Methods Methods of calculus Continuity considerations Algebra Polynomials Quadratic polynomials Investigating the quadratic polynomial

13–15 2.0

Is it true that, if $b > a + c > 0$ , then the quadratic equation $a x^{2} + b x + c = 0$ has two roots?

Problem #PRU-35575

Problems Methods Methods of calculus Continuity considerations Geometry Plane geometry Convex and non-convex figures Convex polygons Calculus Functions of one variable. Continuity Intermediate value theorem. Connectedness.

14–16 5.0

A convex figure and point $A$ inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point $A$ and dividing it in half at point $A$ .

Problem #PRU-78043

Problems Methods Methods of calculus Monotonicity and boundedness Pigeonhole principle Pigeonhole principle (other) Algebra Polynomials Quadratic polynomials

13–15 4.0

The expression $a x^{2} + b x + c$ is an exact fourth power for all integers $x$ . Prove that $a = b = 0$ .