VTMO 2026 - Grade 9 - Final - SET 10 - Self-study

A car must travel 80 km within its planned time. Because of rain, it travels the first quarter of the distance at 15 km/h below the planned speed, then the remaining distance at 10 km/h above the planned speed. Find the planned travel time.

Triangle $ABC$ is right-angled at $A$, with $AB\lt AC$ and altitude $AH$. Let $I,J$ be the incenters of triangles $AHB,AHC$, respectively. Set $D=AI\cap BH$, $F=IJ\cap AC$, and $K=DJ\cap BF$. Prove that the circumcircle of triangle $AKD$ is tangent to $BC$.

a) Given real numbers $a,b$ satisfying $a^3+b^3+6ab=8$, find all possible values of $a+b$.

b) Find all real $a$ such that both $a+3\sqrt{11}$ and $\dfrac1a-3\sqrt{11}$ are integers.

Find all natural numbers $n$ for which $A=n^5+n+1$ is prime.

Let $(O)$ be the circle with diameter $AB$, and let $C$ be any point on segment $AB$. Construct circles $(O_1),(O_2)$ with diameters $AC,CB$. Let $M$ lie on $(O)$, distinct from $A,B$, and let $ME,MF$ be tangents to $(O_1),(O_2)$, touching at $E,F$.

a) Prove $\left(\dfrac{ME}{MA}\right)^2+\left(\dfrac{MF}{MB}\right)^2=1$.

b) Find $C$ such that $ME=\dfrac12MA$.