VTMO 2026 - Grade 9 - Semifinal - SET 09 - Self-study

For how many integers n is $n^2-20n+119$ prime?

Find the area of the triangle formed by $y=3$, $y=2x+1$, and $y=-2x+1$.

For the parabola $y=ax^2$, a≠0, a line through A(2,0) intersects it at two distinct points with abscissas x1, x2. Find $S=1/x_1+1/x_2$.

How many four-digit numbers have no zero digits, end in 9, and are divisible by every one of their digits?

A field contains cows and ducks. An counts 200 heads and Bình 522 legs. How many cows and ducks are there?

Both stores sell notebooks at 10000 dong each. A gives one free for every five paid notebooks; B gives 15% off when buying at least four. What is the minimum cost for 14 notebooks, allowing purchases from both stores?

Circle (O) has diameter AB. D is the midpoint of a semicircular arc AB, and M lies on AB extended outside the circle. MD meets the circle again at C. A circle (I) through D, C is tangent to AB at K; DK meets (O) again at E. Prove that ME is tangent to (O).

Acute triangle ABC with AB$<$AC is inscribed in (O; R). The tangents at B, C meet at M. AM meets (O) again at E. The line through E parallel to BC meets (O) again at F. Circle (I) passes through A and is tangent to BC at C; it meets AF again at N. Prove that BC is the perpendicular bisector of NE.

Ten distinct positive integers sum to 100. What is the largest possible value of the greatest one?

In the pictured 3×3 grid, all 2×2 subsquares have the same sum. Find x.

In the diagram A, B, C, D are concyclic and P is the external intersection of lines AB and DC. Given angles ADB=54°, ACD=58°, PBC=80°, find angle APD.

A grasshopper starts at (0,0). Each jump ends at an integer-coordinate point and has length at most the square root of 5. What is the minimum number of jumps to reach (79,97)?

Solve $(\sqrt{x+24}+\sqrt x)(\sqrt{x+15}-\sqrt{x+3})=12$.

How many integers belong to the domain of $y=\sqrt{x+8}+\dfrac1{x\sqrt{9-x}}$?

How many three-digit even numbers have pairwise distinct digits?

The system $x+2y=4$, $ax+by=-8$ has infinitely many solutions. Find a+b.

Ten equations $x^2+a_ix+b_i=0$ have 20 distinct roots in total, namely ±1,±2,...,±10. Let M be the minimum possible sum of the b_i. Find the final digit of |M|.

Two concentric circles centered at O have radii 1 and 2. OM=3. The two tangents from M to the inner circle meet the outer circle at A, B and C, D, with MA$<$MB and MC$<$MD. Find BD.

Arrange six distinct cars in a row: one green, two yellow and three red. How many arrangements have no adjacent cars of the same color?

Let a, b be the largest and smallest integer values of m for which $x^2+(2m+1)x+m^2-8m+7=0$ has roots of opposite signs. Find a−b.