VTMO 2026 - Grade 9 - Final - SET 11 - Self-study

Let $(O)$ be the circle with diameter $AB$, and let $M$ be the midpoint of $OB$. The line through $M$ perpendicular to $AB$ meets the circle at $C$. Choose $E$ on $AB$, distinct from $A,B$, and let $CE$ meet the circle again at $D$. Let $H$ be the foot of the perpendicular from $B$ to $CD$.

a) Prove $CH=\dfrac12AD$.

b) Let $BH$ meet the circle again at $F$, and let $I$ be the midpoint of $DF$. Prove that $I$ lies on a fixed circle as $E$ varies.

Let nonnegative real numbers $x,y,z$ satisfy
$$x^2+y^2=1,\qquad y^2+z^2=2,\qquad xz+y^2\ge\sqrt2.$$
Find the minimum of $M=x+z$.

Let $a,b,c$ be pairwise distinct real numbers. Find the minimum of
$$P=\frac{(a-b)^4}{(b-c)^2(c-a)^2}+\frac{(b-c)^4}{(c-a)^2(a-b)^2}+\frac{(c-a)^4}{(a-b)^2(b-c)^2}.$$

Let $M$ consist of 2024 positive numbers $a_1,a_2,\ldots,a_{2024}$. For every nonempty subset $T_i$ of $M$, let $s_i$ be the sum of its elements. Prove that all the resulting sums can be partitioned into 2024 nonempty disjoint groups such that the ratio of any two numbers in the same group is at most 2.

A natural number is called super-even if all its digits are even. How many four-digit super-even numbers remain super-even when multiplied by 3?