Walking into a math competition without the right formulas is like showing up to a cooking contest without knowing how to boil water. You might have talent and intuition, but you’ll waste precious minutes reinventing wheels that other competitors already have spinning. For SASMO participants, knowing which formulas to memorize and when to apply them can mean the difference between a bronze medal and a gold.

Core arithmetic formulas that save time

Mental math speed separates good SASMO performers from great ones. The competition rewards students who can calculate accurately without reaching for scratch paper every ten seconds.

Start with percentage conversions. Memorize that 12.5% equals one eighth, 16.67% equals one sixth, and 33.33% equals one third. When a problem asks for 37.5% of 240, students who recognize three eighths can multiply 240 by 3 and divide by 8 in their heads. That’s ten seconds versus forty.

Fraction to decimal conversions matter too. Know that one seventh cycles as 0.142857, one ninth becomes 0.111…, and one eleventh alternates digits. These patterns appear in ratio problems disguised as decimals.

Square numbers through 25 squared should be automatic. Cube numbers through 10 cubed help with volume calculations. Perfect squares let you spot factorization opportunities that other students miss.

The 5 mental math shortcuts every SASMO competitor should master can reduce calculation time by half once they become second nature.

Number theory formulas every competitor needs

Number theory questions show up in nearly every SASMO paper. Students who know divisibility rules and factor patterns spot solutions while others are still setting up long division.

Divisibility rules worth memorizing:

• A number divides by 3 if its digit sum divides by 3
• A number divides by 4 if its last two digits divide by 4
• A number divides by 6 if it’s even and divisible by 3
• A number divides by 8 if its last three digits divide by 8
• A number divides by 9 if its digit sum divides by 9
• A number divides by 11 if the alternating digit sum divides by 11

The sum of divisors formula helps with factor problems. For a number n = p^a × q^b, the sum of divisors equals [(p^(a+1) – 1)/(p – 1)] × [(q^(b+1) – 1)/(q – 1)].

The number of divisors formula is simpler. For n = p^a × q^b × r^c, the divisor count equals (a+1)(b+1)(c+1).

Prime factorization unlocks most number theory problems. Students should know primes through 100 by heart. When you see 143, you should immediately think 11 × 13.

Understanding why number theory is the secret weapon every SASMO competitor needs transforms how students approach these problems.

Geometry formulas that appear repeatedly

Geometry questions test whether students know their formulas cold or waste time deriving basics from scratch.

Shape	Area Formula	Perimeter Formula
Triangle	(base × height) ÷ 2	sum of all sides
Triangle (Heron’s)	√[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2	a + b + c
Parallelogram	base × height	2(length + width)
Trapezoid	[(base₁ + base₂) × height] ÷ 2	sum of all sides
Circle	πr²	2πr
Sector	(θ/360) × πr²	2r + (θ/360) × 2πr

The Pythagorean theorem (a² + b² = c²) appears in half of all geometry problems. Memorize common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. When you see these numbers, you know you’re dealing with a right triangle.

Special right triangles save calculation time. The 30-60-90 triangle has sides in ratio 1 : √3 : 2. The 45-45-90 triangle has sides in ratio 1 : 1 : √2.

Circle theorems matter for upper primary levels. The angle in a semicircle equals 90 degrees. Angles at the circumference subtended by the same arc are equal. The angle at the center is twice the angle at the circumference.

Volume formulas for 3D shapes:

1. Cube: side³
2. Rectangular prism: length × width × height
3. Cylinder: πr²h
4. Cone: (πr²h) ÷ 3
5. Sphere: (4πr³) ÷ 3

Students preparing for geometry sections should review 7 geometry theorems that appear in nearly every SASMO paper to see how these formulas connect.

Algebra shortcuts that unlock problems

Algebraic thinking separates primary students who can handle abstract reasoning from those still tied to concrete numbers.

The difference of squares formula (a² – b²) = (a + b)(a – b) appears constantly. When you see 47² – 43², don’t calculate each square. Recognize (47 + 43)(47 – 43) = 90 × 4 = 360.

Perfect square trinomials follow two patterns:
* (a + b)² = a² + 2ab + b²
* (a – b)² = a² – 2ab + b²

Sum and product of roots for quadratic equations help with coefficient problems. For ax² + bx + c = 0, the sum of roots equals -b/a and the product equals c/a.

Arithmetic sequence formulas:
* nth term: a + (n-1)d
* Sum of n terms: [n(first term + last term)] ÷ 2

Geometric sequence formulas:
* nth term: ar^(n-1)
* Sum of n terms: a(r^n – 1) ÷ (r – 1)

The formula for the sum of first n natural numbers is n(n+1) ÷ 2. For the sum of first n squares, use n(n+1)(2n+1) ÷ 6.

Building how to build strong algebraic thinking for math olympiads requires understanding when to apply each formula type.

Combinatorics formulas for counting problems

Counting problems trip up students who don’t know whether to multiply, add, or use combinations. The formulas themselves are simple. Knowing which one to use is the challenge.

Permutations (order matters): nPr = n! ÷ (n-r)!

Combinations (order doesn’t matter): nCr = n! ÷ [r!(n-r)!]

The fundamental counting principle states that if one event can occur in m ways and another in n ways, together they can occur in m × n ways.

Circular permutation formula: (n-1)! for arranging n objects in a circle.

The inclusion-exclusion principle: |A ∪ B| = |A| + |B| – |A ∩ B|

For three sets: |A ∪ B ∪ C| = |A| + |B| + |C| – |A ∩ B| – |A ∩ C| – |B ∩ C| + |A ∩ B ∩ C|

The biggest mistake students make with counting formulas is not drawing a diagram first. A simple tree diagram or organized list often reveals the pattern faster than jumping straight to formulas. Use formulas to verify your answer, not to replace understanding.

Students who want deeper understanding should check out combinatorics made simple: counting principles for SASMO success.

Ratio and proportion formulas

Ratio problems appear in every SASMO level. The formulas are straightforward, but application requires careful attention to what the problem actually asks.

Basic ratio formula: If a:b = c:d, then ad = bc (cross multiplication).

When three quantities are in ratio a:b:c and their sum is S, each part equals:
* First part: (a × S) ÷ (a + b + c)
* Second part: (b × S) ÷ (a + b + c)
* Third part: (c × S) ÷ (a + b + c)

Direct proportion: y = kx (as one increases, the other increases proportionally)

Inverse proportion: y = k/x (as one increases, the other decreases proportionally)

Speed, distance, and time relationships:
* Distance = Speed × Time
* Speed = Distance ÷ Time
* Time = Distance ÷ Speed

Work rate problems use the formula: Work = Rate × Time. When multiple workers combine efforts, add their rates.

Mixture problems use the weighted average formula: Final concentration = (Amount₁ × Concentration₁ + Amount₂ × Concentration₂) ÷ Total Amount

For comprehensive coverage of these concepts, review the complete guide to ratio and proportion problems in SASMO.

Common formula application mistakes

Knowing formulas and applying them correctly are two different skills. Here are the mistakes that cost students points.

Mistake 1: Using the wrong formula for the situation

Students see a triangle and immediately use base × height ÷ 2, even when the problem gives three sides and no height. That’s when Heron’s formula is needed.

Mistake 2: Forgetting to convert units

A problem gives radius in centimeters and asks for area in square meters. Students calculate correctly but forget to convert, losing the entire problem.

Mistake 3: Mixing up permutation and combination

If the problem asks how many ways to choose 3 students from 10, that’s a combination. If it asks how many ways to arrange them in order, that’s a permutation.

Mistake 4: Applying formulas without understanding context

A student memorizes that the sum of angles in a polygon is (n-2) × 180° but doesn’t realize that formula gives the total, not each individual angle.

Mistake 5: Rounding too early

Using π = 3 or √2 = 1.4 in intermediate steps compounds errors. Keep full precision until the final answer.

Building your formula reference sheet

Creating a personal formula sheet helps solidify memory and provides a competition day study tool.

Start by organizing formulas by topic: arithmetic, number theory, geometry, algebra, combinatorics, and ratios. Write each formula clearly with its conditions and limitations.

Add example problems next to each formula. Seeing “a² – b² = (a+b)(a-b)” with “Example: 99² – 1² = (99+1)(99-1) = 100 × 98 = 9800” makes the formula stick.

Include common variations. For area of a triangle, list all methods: base × height ÷ 2, Heron’s formula, and (ab sin C) ÷ 2.

Highlight formulas that appear most frequently. After working through past papers, you’ll notice certain formulas show up repeatedly. Mark these with a star or different color.

Review your sheet weekly, not just the day before competition. Spaced repetition beats cramming every time.

Understanding what makes a problem solvable: understanding mathematical logic in competitions helps you know which formula to reach for when.

Practice strategies for formula mastery

Memorizing formulas without practice is like learning vocabulary without reading. You need context to make knowledge stick.

Work through past SASMO papers systematically. When you encounter a problem, identify which formula applies before calculating. This builds pattern recognition.

Time yourself on formula-heavy sections. Set a goal to complete basic calculations in under 30 seconds. Speed comes from automatic recall, not from thinking through derivations.

Create formula flashcards with the formula on one side and a sample problem on the other. Quiz yourself daily during breakfast or on the way to school.

Teach the formulas to someone else. Explaining why the area of a circle is πr² and not 2πr (that’s circumference) to a younger sibling or parent reinforces your own understanding.

Join study groups where students challenge each other with formula application problems. Peer teaching catches gaps that solo study misses.

Before competition day, review how to manage your time effectively during SASMO competition day to ensure your formula knowledge translates to points.

Turning memorization into understanding

The best SASMO competitors don’t just memorize formulas. They understand why formulas work and when to modify them.

Take the area of a trapezoid: [(base₁ + base₂) × height] ÷ 2. This formula makes sense when you realize you’re finding the average of the two bases and multiplying by height, just like a rectangle.

The combination formula nCr = n! ÷ [r!(n-r)!] becomes intuitive when you understand you’re counting permutations (n!) then dividing out the arrangements that don’t matter (r! for chosen items, (n-r)! for unchosen items).

Understanding formula derivations helps when you forget the exact form under pressure. If you blank on Heron’s formula but remember it comes from the Pythagorean theorem and altitude relationships, you can reconstruct it.

Connect formulas across topics. The sum of an arithmetic sequence relates to the area of a trapezoid. Both involve averaging the first and last terms. These connections make formulas easier to remember and apply creatively.

Your formula foundation starts now

SASMO competition preparation formulas aren’t just facts to memorize. They’re tools that unlock problems other students find impossible.

Start building your formula sheet today. Pick one topic and master those formulas completely before moving to the next. Practice until you can write each formula from memory and explain what each variable represents.

The students who medal at SASMO aren’t necessarily smarter. They’ve just done the work to make essential formulas automatic, freeing their minds to focus on problem solving strategy instead of basic calculations. That advantage compounds over every question on test day.

Your formula mastery journey begins with a single sheet of paper and fifteen minutes of focused practice. The medals come later.