Question 1
Santa's basket contains three kinds of candy: chocolate, orange and aloe vera. Every selection of 100 candies contains all three kinds. What is the greatest possible total number of candies in the basket?
VTMO 2026 - Grade 6 - Final - SET 5 - Self-study
Question 2
Pi thinks of a five-digit positive integer and deletes one digit to obtain a four-digit integer. The sum of the original and resulting numbers is 52713. Find the original five-digit number.
Question 3
Teams $A$ and $B$ each have 10 workers. Team $A$ can build a road in 14 days and a bridge in 25 days. Team $B$ can build the same road in 18 days and the same bridge in 15 days. Team $C$ consists of five workers from $A$ and five from $B$. It builds the road first and then the bridge. How many days does it need altogether?
Model clarification: for each job, all workers within an original team have equal, constant productivity, and their work rates add. Productivity may differ between the two teams and between the two jobs.
Model clarification: for each job, all workers within an original team have equal, constant productivity, and their work rates add. Productivity may differ between the two teams and between the two jobs.
Question 4
a) Find the six-digit natural number $N$ formed only from digits 8 and 9 that is divisible by $2^6$.
b) Use digits $1,2,3,\ldots,9$ exactly once altogether to form three three-digit numbers $a,b,c$ such that $a:b:c=1:3:5$. Find $a,b,c$.
b) Use digits $1,2,3,\ldots,9$ exactly once altogether to form three three-digit numbers $a,b,c$ such that $a:b:c=1:3:5$. Find $a,b,c$.
Question 5
A number is called a palindrome if it reads the same from left to right and from right to left; examples include $7,11,121,1221,\ldots$. Let $\overline{abccba}$ be a six-digit palindrome.
a) Prove that $\overline{abccba}$ is divisible by 11.
b) How many such six-digit palindromes are divisible by 91?
a) Prove that $\overline{abccba}$ is divisible by 11.
b) How many such six-digit palindromes are divisible by 91?
Question 6
In a competition, 25% of all contestants receive prizes. Of the boys, $2/7$ receive prizes and 420 do not. There are 648 girls in the competition. How many girls receive prizes?
Question 7
Rectangle $ABCD$ has $AD=18$ cm and $AB=12$ cm. Points $E,F$ lie on $AD,BC$, respectively, with $AE=BF=6$ cm. Diagonals $AC,BD$ meet at $O$ and meet $EF$ at $P,Q$, respectively. Find the total area of the shaded regions in the diagram, in cm².
Question 8
A large cube is built from unit cubes. Some of its outer faces are painted completely. After the cube is taken apart, exactly 45 unit cubes have no paint. How many unit-cube faces have been painted?
Question 9
a) The first diagram combines a triangle, a circle and a square of equal area. The shaded regions are the overlaps shown. The unshaded part of the triangle is $5/8$ of the whole triangle, and half the circle is shaded. Find the ratio of the unshaded area to the area of the entire combined figure.
b) In the second diagram, triangle $PQR$ has all three sides equal to 3 cm. Points $U,V,W,X,Y,Z$ divide the sides into 1 cm segments as shown. Find the ratio of the shaded area to the area of triangle $PQR$.
Label clarification: $W$, the left division point on $QR$, is included explicitly in the list of marked points.
b) In the second diagram, triangle $PQR$ has all three sides equal to 3 cm. Points $U,V,W,X,Y,Z$ divide the sides into 1 cm segments as shown. Find the ratio of the shaded area to the area of triangle $PQR$.
Label clarification: $W$, the left division point on $QR$, is included explicitly in the list of marked points.
Question 10
The set $A=\{1,2\}$ has four subsets: $\varnothing,\{1\},\{2\},\{1,2\}$. Their element sums are $0,1,2,3$, respectively. Let $S[A]$ denote the sum of these sums over all subsets of $A$. Thus $S[A]=0+1+2+3=6$.
a) For $B=\{1,2,3\}$, find $S[B]$.
b) For $C=\{a,b,c,d\}$, where $a,b,c,d$ are distinct natural numbers, prove that $S[C]$ is divisible by 8.
a) For $B=\{1,2,3\}$, find $S[B]$.
b) For $C=\{a,b,c,d\}$, where $a,b,c,d$ are distinct natural numbers, prove that $S[C]$ is divisible by 8.
Question 11
A natural number is called a “Harshad number” if it is divisible by the sum of all its digits. For example, $2024$ is a Harshad number because $2+0+2+4=8$ and $2024$ is divisible by $8$. How many four-digit Harshad numbers are less than $2024$ and divisible by $9$?
Question 12
Find all pairs of positive integers $a,b$ satisfying both conditions: $a,a+2b,a+4b,a+8b,a+16b$ are prime numbers; and $b,b+2a,b+4a,b+8a,b+16a$ are prime numbers. Give a complete proof.
Question 13
The first car leaves city $A$ for city $B$ and immediately turns back towards $A$ on reaching $B$. The second car leaves $B$ for $A$ and immediately turns back towards $B$ on reaching $A$. They start at the same time. Their first meeting is $30$ km from $A$. Their second meeting is $72$ minutes later and $15$ km from $B$; at that moment the first car is returning towards $A$ after reaching $B$. Find the speed of each car.
Practice-model clarification: each car travels at its own constant speed throughout, and before their second meeting both cars have reached the opposite city and turned back. The diagram is schematic, not to scale.
Practice-model clarification: each car travels at its own constant speed throughout, and before their second meeting both cars have reached the opposite city and turned back. The diagram is schematic, not to scale.