VTMO 2026 - Grade 9 - Semifinal - SET 04 - Self-study

In the coordinate plane, let $(d_1):y=ax+3$ and $(d_2):y=bx+3$. Let $A=d_1\cap d_2$, $B=d_1\cap Ox$, and $C=d_2\cap Ox$. Triangle $ABC$ is right-angled at $A$ and has a $60^\circ$ angle. Find $P=a^2+b^2$.

Let $(O)$ be a circle with diameter $AB$ and let $C$ lie on the circle. The line through $C$ parallel to $AB$ meets the circle at $D$. Choose $E$ on $AB$ with $AE=BD$. Extend $CE$ to meet the circle at $M$, and extend $DO$ to meet it at $N$. Let $F=AN\cap BM$. Prove that $EF\perp AB$.

How many triangles with vertices chosen from a regular 20-gon have an angle greater than $100^\circ$?

An and Bình complete a job together in three hours. Their individual completion times in hours are natural numbers, and Bình works faster than An. How many hours would Bình need alone?

Let $M=n^2+n+37$, where $n$ is a natural number. For how many $n$ from 1 to 50 is $M$ a power of 7?

Let $n$ be even with 15 positive divisors. What is the maximum number of positive divisors of $3n/2$?

How many three-digit natural numbers have adjacent digits differing by at most 1? Examples include 123,434,987,221,555.

The points in the coordinate plane are shown in the figure. Triangles $ADB$ and $CBO$ have equal areas. Find $a$.

Let $a,b$ be integers such that $x^2+ax+1-b=0$ has two nonzero integer roots. Prove that $a^2+b^2$ is composite.

Let positive integers $a,b,c,d$ satisfy $a^3=b^2$, $c^5=d^4$, and $a-c=9$. Find $b-d$.

How many positive integer pairs $(x,y)$ satisfy $x\lt y$, $x+y=2020$, and $20\mid xy$?

Let acute triangle $ABC$ be isosceles at $A$ and inscribed in $(O)$. Choose $D$ on minor arc $BC$ with $DB\lt DC$. Lines through $B,C$ parallel to $AD$ meet the circle again at $E,F$. Choose $K$ on minor arc $AE$, distinct from $A,E$. Let $M=KE\cap AC$ and $N=KF\cap AB$. Prove that $MN,KD,BC$ are concurrent.

An starts 100 m west of Bình and Châu 160 m east of Bình. An walks south at 20 m/min and Châu north at 41 m/min, starting together, while Bình stays still. When is Châu's distance from Bình twice An's distance from Bình?

Consider
$$A=37^n+71^n-2^n-1,$$
$$B=n(n^2-6)^2(n^2-1)-n^3(n-1)(n+1),$$
$$C=1+2+\cdots+(21n+1).$$
Which expressions are divisible by 21 for every natural number $n$?

Eight friends plan to split a restaurant bill equally. Tuấn forgets his money, so each of the other seven pays an extra 25,000 VND. What is the total bill?

An drives at an average speed of 55 km/h for an integer number of hours. The odometer initially reads $\overline{abc}$, with $a\ge1$ and $a+b+c\le7$, and finally reads $\overline{cba}$. Find $a^2+b^2+c^2$.

How many five-digit numbers with distinct digits can be formed from $0,1,2,3,4,5$?

A date is prime if both its day number and month number are prime. For example, March 29 is prime but March 28 is not. How many prime dates occur in 2019?

Let $x_1,x_2$ be the roots of $(x+1)(2x-3)+(x-4)(x+11)+38=0$. Find $x_1^2+x_2^2$.

Find the sum of all real solutions of $(x+4\sqrt{x}+3)(x+12\sqrt{x}+27)=320x$.