VTMO 2026 - Grade 9 - Semifinal - SET 08 - 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 p, q be distinct primes. Put $N=p^3q^4$ and $A=\{n/N:1\le n\le N,\ n\in\mathbb N,\ \gcd(n,pq)=1\}$. Find the sum of all elements of A.

A circle is divided into 12 equal arcs by points A through L in order. Angles x and y are marked as shown. Find $x+y$.

Let the isosceles trapezoid $ABCD$, with $AB\parallel CD$ and $AB\lt CD$, be inscribed in a circle with centre $O$. Let $K$ be the intersection of its diagonals. Let $M,N$ be the circumcentres of triangles $KAB,KCD$, respectively, and let $P,Q$ be the circumcentres of $KAD,KBC$, respectively.

a) Prove that $MQNP$ is a parallelogram.

b) Prove that $KM=ON$.

Fold the pictured net into a cube, with one number on each face. Find the largest product of the three numbers on faces meeting at a vertex.

Ten pigs are weighed one at a time. After each weighing, the average weight of all pigs weighed so far is calculated; successive averages increase by 1 kg. How much heavier is the tenth pig than the first?

Which segment in the diagram is the reflection of PQ across the x-axis?

There are 60 chairs around a round table. Students sit so that each student has exactly one occupied neighboring chair. What is the maximum number of students?

Fill a $5\times5$ grid with 1,2,3 so that every $2\times2$ square contains all three numbers. Find the minimum total S.

Four equal circles are packed inside triangle ABC, right-angled at A, with the tangencies shown. Given AB=27 and AC=36, find their common radius.

Hạnh and Phúc must fold 24 gliders and 15 cranes. Hạnh takes 2 minutes per glider and 6 per crane; Phúc takes 3 per glider and 5 per crane. They work simultaneously. What is the minimum completion time in minutes?

How many polynomials $f(x)=x^2-ax-b$, with positive integers a, b, have at least one positive root smaller than 4?

A, B, C, D have 76 candies in total. Simultaneously A keeps 1, B keeps 2, C keeps 3 and D keeps 4; each divides the remainder equally among the other three. Afterwards all four have equal amounts. How many candies did A initially have?

Positive integers a, b satisfy $\dfrac1a+\dfrac2b=\dfrac8{2a+b}$ and $2010\le4a+7b\le2016$. Find a+b.

Natural numbers a, b have remainders 1 and 2 when divided by 8. What is the remainder of $a^2-b^2$ modulo 8?

BA=6 cm is tangent to a circle at A. A secant from B meets the circle at C and then D as shown, with CD=9 cm. Find BC.

In a circle with diameter AB, C lies on AB with BC=2 AC. D lies on the circle and DC is perpendicular to AB; DE is another diameter as shown. Find the area ratio of triangles CDE and ADB.

Cyclic quadrilateral ABCD has diagonals AC and BD perpendicular at I, with IA=3, IB=4, ID=6 cm as shown. Find CD.

On segment AB of length 30m, three bugs start simultaneously: bug 1 moves from A to B at 2m/min; bug 2 starts 10m from A and moves toward B at 0.5m/min; bug 3 moves from B toward A at 0.5m/min. They stop at an endpoint. Find the minimum AC, with C on AB, such that at every instant from their simultaneous start until all three have completely stopped, at least one bug is crawling on segment AC.

Five football teams play a single round robin. A win gives 3 points, a loss 0, a draw 1 to each team. After some games all five scores are positive and pairwise distinct. What is the minimum number of games played?