VTMO 2026 - Grade 7 - Final - SET 8 - Self-study

Given six irrational numbers, prove that three of them, $a,b,c$, can be chosen so that $a+b,b+c,c+a$ are all irrational.

Find the minimum value of $C=\dfrac6{|x|-3}$ for integer $x$.

Let $M=\overline{abcdefgh}$ be an eight-digit number using each of $1,2,3,4,5,6,7,8$ once. Its two-digit prefix is divisible by 2, its three-digit prefix by 3, and so on up to its eight-digit prefix being divisible by 8. Find $M$.

a) Equal segments $AB$ and $CD$ intersect at $K$, and $AC\parallel BD$. Prove that $KA=KC$.

b) Triangle $ABC$ is isosceles at $A$. Its angle bisector $BD$ meets $AC$ at $D$. Altitude $AH$ satisfies $AH=\dfrac12BD$. Find $\angle BAC$.

Four players $A,B,C,D$ each wear a shirt numbered 6, 7, or 8. A player wearing 7 always lies; a player wearing 6 or 8 always tells the truth. Asked for the sum of all four shirt numbers, they reply:

$A$: “The sum is 28.”

$B$: “The sum is 27.”

$C$: “The sum is 26.”

$D$: “The sum is 25.”

Find every player’s shirt number. Give the ordered tuple $(A,B,C,D)$.