Designed by Mr.Shen

UOJMC Training | Lesson 2

Competition preparation: Number Theory, Algebra, and Spatial Geometry.

Number Theory & Algebra

Competition questions love to test your ability to turn English sentences into algebraic equations and manipulate prime numbers.

Scenario A: Even/Odd Consecutive Integers

If a question asks for consecutive EVEN or ODD numbers, they jump by twos! The setup becomes: $x$, then $x + 2$, then $x + 4$.

Problem: The sum of three consecutive EVEN numbers is 126. Find the largest number.
Step 1: Set up variables: $x$, $x+2$, $x+4$.
Step 2: Build equation: $x + (x+2) + (x+4) = 126$
Step 3: Simplify: $3x + 6 = 126$
Step 4: Solve: $3x = 120 \rightarrow x = 40$
Largest is $(x+4)$: $40 + 4 = 44$.
Scenario B: Prime Factorization (Index Form)

Instead of listing factors, we use prime index form to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM).

  • HCF: Take the lowest power of common primes.
  • LCM: Take the highest power of all primes present.
Problem: Find HCF and LCM of $A = 2^3 \times 3^2 \times 5$ and $B = 2^2 \times 3^3 \times 7$
HCF: Common primes are 2 and 3.
Lowest powers: $2^2$ and $3^2$.
$\rightarrow \text{HCF} = 2^2 \times 3^2$
LCM: Primes present: 2, 3, 5, 7.
Highest powers: $2^3$, $3^3$, $5^1$, $7^1$.
$\rightarrow \text{LCM} = 2^3 \times 3^3 \times 5 \times 7$
Scenario C: Number Patterns

Some sequences aren't linear (like $2, 4, 6...$). The most common non-linear pattern in UOJMC is Triangular Numbers.

They are called triangular because they form equilateral triangles when drawn as dots. The sequence goes: 1, 3, 6, 10, 15...

Notice the gap increases by 1 each time! (+2, +3, +4...).

Knowledge Check

1. Number $X = 2^3 \times 3^2$ and Number $Y = 2^2 \times 3^3$. Calculate the exact value of their Highest Common Factor (HCF).

2. The sum of THREE consecutive ODD integers is 57. What is the LARGEST of these integers?

3. Using the triangular number pattern, what is the 6th triangular number?

Solutions:
1. Lowest powers are $2^2$ and $3^2$. $4 \times 9 =$ 36.
2. Set up algebra: $x + (x+2) + (x+4) = 57 \Rightarrow 3x + 6 = 57 \Rightarrow 3x = 51 \Rightarrow x = 17$. Largest is $17+4 =$ 21.
3. Pattern adds 2, then 3, then 4, then 5... The 5th is 15. The 6th adds 6: $15 + 6 =$ 21.

Geometry & Space

Visualizing unseen lines is key here. Complex shapes can usually be broken down into simpler right-angled triangles or squares.

Scenario A: Coordinate Distance (Pythagoras)

To find the distance between two points on a grid, draw a right-angled triangle between them. Use Pythagoras: $a^2 + b^2 = c^2$.

Problem: Find the shortest distance between $(1, 2)$ and $(4, 6)$.
Step 1: Find horizontal base: $4 - 1 = 3$ units.
Step 2: Find vertical height: $6 - 2 = 4$ units.
Step 3: Pythagoras: $3^2 + 4^2 = c^2$
$9 + 16 = c^2 \rightarrow 25 = c^2$
Distance ($c$) is $\sqrt{25} = 5$.
Scenario B: Compound 2D Shapes

When shapes are stuck together, perimeter ONLY counts the outside edges. Inner lines vanish.

Problem: A rectangle is made of 3 identical squares. The total area is 75. Find the perimeter.
Step 1: Find area of ONE square: $75 \div 3 = 25$.
Step 2: Find side length of square: $\sqrt{25} = 5$.
Step 3: Count outer edges. The rectangle is 3 squares long (length = 15) and 1 high (height = 5).
Perimeter = $15 + 15 + 5 + 5 = 40$.

Knowledge Check

1. What is the shortest straight-line distance between point $(0,0)$ and point $(5,12)$?

2. A right-angled triangle has a base of 10 and a hypotenuse of 26. What is its Area?

3. A rectangle is made of 4 identical squares in a row. Its total area is 64. What is the PERIMETER?

Solutions:
1. Pythagoras: $5^2 + 12^2 = 25 + 144 = 169$. The square root of 169 is 13.
2. Find height first: $26^2 - 10^2 = 676 - 100 = 576$. $\sqrt{576} = 24$. Area = $\frac{1}{2} \times 10 \times 24 = $ 120.
3. Area of 1 square = $64 \div 4 = 16$. Side length = 4. Rectangle is 16 long, 4 high. Perimeter = $16 + 16 + 4 + 4 =$ 40.

Boss Level: The Painted Cube

3D Spatial Visualization

This is a classic UOJMC question. You must picture a 3-dimensional object being taken apart piece by piece.

The Setup

Take a solid wooden cube measuring 4 units by 4 units by 4 units.

You paint the entire outside of this large cube completely black.

You then use a saw to slice it up into smaller $1 \times 1 \times 1$ unit cubes. (There are $4 \times 4 \times 4 = 64$ unit cubes in total).

The Final Gauntlet

1. How many of the unit cubes have exactly 3 faces painted black?

2. How many of the unit cubes have exactly 2 faces painted black?

3. How many of the unit cubes have NO faces painted black? (The unpainted inner core).

Solutions:
1. 3 faces painted = The Corners. A cube always has exactly 8 corners.
2. 2 faces painted = The Edges. A cube has 12 edges. On a length of 4, the 2 ends are corners, leaving 2 middle pieces per edge. $12 \text{ edges} \times 2 =$ 24.
3. No faces painted = The inner core. Subtract the outer shell (1 unit on all sides) from the dimensions. $4-2 = 2$. The inner core is $2 \times 2 \times 2 =$ 8 cubes.

Homework Circuit

1. Multiples & Factors

Find the SMALLEST number that is strictly greater than 100, which is a perfect multiple of 6, 8, and 12.

Number:

2. Algebraic Equations

The sum of three consecutive ODD integers is exactly 105.

What is the LARGEST of these integers?

3. Logical Sequences

The sequence of triangular numbers is 1, 3, 6, 10...

Calculate the SUM of the 6th and 7th triangular numbers.

4. Compound 2D Shapes

Three completely identical squares are placed side-by-side in a single row to form a rectangle. The total area of this new rectangle is $75 \text{ cm}^2$.

What is the perimeter of this rectangle?

5. Coordinate Geometry

On a standard Cartesian plane, point A is located at coordinate $(0,0)$ and point B is located at coordinate $(6,8)$.

What is the shortest straight-line distance between A and B?

6. Right-Angled Triangles

A right-angled triangle has a base measuring 5 cm and a hypotenuse (the longest side) measuring 13 cm.

What is the area of the triangle?

7. The Painted Cube (Advanced)

A solid wooden cube measuring $5 \times 5 \times 5$ is painted completely blue on all outer faces. It is then cut into smaller $1 \times 1 \times 1$ unit cubes.

a) How many unit cubes have exactly 3 blue faces?
b) How many unit cubes have exactly 2 blue faces?
c) How many unit cubes are completely unpainted?

Marking Schedule

Score: / 9
1. Multiples & Factors
Step 1: Find the Lowest Common Multiple (LCM) of 6, 8, and 12. Using prime factors, the LCM is 24.
Step 2: List the multiples of 24: 24, 48, 72, 96, 120...
The smallest multiple strictly greater than 100 is 120.
2. Algebraic Equations
Step 1: Let the three consecutive odd integers be $x$, $x+2$, and $x+4$.
Step 2: Set up the equation: $x + (x+2) + (x+4) = 105$.
Step 3: Simplify to $3x + 6 = 105$, which means $3x = 99$, so $x = 33$.
The largest integer is $x+4 \rightarrow 33 + 4 =$ 37.
3. Logical Sequences
Step 1: Map the sequence and its gaps: 1 (+2) 3 (+3) 6 (+4) 10 (+5) 15 (+6) 21 (+7) 28.
Step 2: Identify the 6th number (21) and the 7th number (28).
Calculate the sum: $21 + 28 =$ 49.
4. Compound 2D Shapes
Step 1: Find the area of ONE square: $75 \div 3 = 25 \text{ cm}^2$.
Step 2: Find the side length of the square: $\sqrt{25} = 5 \text{ cm}$.
Step 3: The rectangle is 3 squares long ($15 \text{ cm}$) and 1 square high ($5 \text{ cm}$).
Perimeter = $15 + 15 + 5 + 5 =$ 40.
5. Coordinate Geometry
Step 1: Imagine a right triangle. The horizontal base is $6-0=6$. The vertical height is $8-0=8$.
Step 2: Apply Pythagoras: $6^2 + 8^2 = c^2$.
Step 3: $36 + 64 = 100$. So, $c^2 = 100$.
Distance ($c$) is $\sqrt{100} =$ 10.
6. Right-Angled Triangles
Step 1: Find the missing height using Pythagoras: $5^2 + h^2 = 13^2$.
Step 2: $25 + h^2 = 169 \rightarrow h^2 = 144 \rightarrow h = 12$.
Step 3: Apply Area formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
$\text{Area} = \frac{1}{2} \times 5 \times 12 =$ 30.
7a. The Painted Cube (3 Faces)
Step 1: Unit cubes with exactly 3 painted faces are always the corner pieces of the larger cube.
A cube always has exactly 8 corners. Answer: 8.
7b & 7c. Painted Cube (2 & 0 Faces)
2 Faces: These are the edge pieces. A cube has 12 edges. On a 5-unit edge, the 2 ends are corners, leaving 3 middle pieces. $12 \text{ edges} \times 3 =$ 36.
0 Faces: This is the unpainted inner core. Subtract the outer painted shell (1 unit on all sides) from the overall dimensions: $5 - 2 = 3$.
The inner core is a $3 \times 3 \times 3$ cube. $3 \times 3 \times 3 =$ 27.