This is a sample of competition-level math problems that are used to illustrate KQ methods.
| Front | Evaluate \(\lfloor\frac{n}{2}\rfloor + \lfloor\frac{n+1}{2}\rfloor\). |
| Back | Stall: Casework errors.KQ unlocker: Try even/odd once \(\rightarrow\) always = n.KQ hint: "Test n=2k and n=2k+1."Transfer: Many parity/floor identities. |
| Front | Solve |2x-5|=|x+1|. |
| Back | Stall: Sign mistakes.KQ unlocker: Cases around x=\(\frac{5}{2}\) and x=-1; or square both sides after noting both nonnegative.KQ hint: "Break at where insides change sign."Transfer: Absolute value equations/ineqs. |
| Front | Given \(f(x) = 2f(x-1) + 1, f(0)=3,\) find \(f(3).\) |
| Back | Stall: Unsystematic plugging.KQ unlocker: Buid table recursively: \(f(1)=7, f(2)=15, f(3)=31\) (or see \(f(n)=2^{n+2}-1\)).KQ hint: "Compute a few; what the pattern?"Transfer: Linear recurrences, binary growth. |