The MOEMS packet comes home from school with five dates circled, November through March, and nothing to actually practice with in between.
Search for MOEMS practice problems and you’ll mostly find two things: an official page of sample questions with zero explanation, or a $30 book that assumes you already know which division fits. Neither helps tonight.
Below are ten original MOEMS-style problems, five for Division E and five for Division M, each worked out step by step, built specifically to fill that in-between-contests gap.
If you still need the basics, what the divisions mean, how scoring works, when to register, our MOEMS parent’s guide to divisions, format and prep covers all of that; this page is just the practice.
Why We Wrote Original MOEMS Practice Problems Instead of Reposting Old Contests
MOEMS problems are the organization’s copyrighted material, sold as “Contest Problems” volumes through its store and on Amazon. That’s part of why the search results for this term are so thin: real past contest problems are either paywalled or turning up on ad-heavy PDF sites that don’t say where they got them.
We checked our own Search Console data before writing this page. Over the past 90 days, gonit.app got 64 impressions, and exactly 1 click across 16 MOEMS-related searches, including one for “moems contest problems pdf.” Small numbers, but a real, mostly-unanswered pattern of parents typing this exact question into Google. That’s the honest reason this page exists: not a huge keyword, just a real one nobody had answered well yet.
The one resource we’d actually recommend is moems.org’s own sample tournament questions page: official, free, both divisions. It doesn’t explain anything though, which is exactly the gap this page fills. Everything below is original, written in the MOEMS style for the kinds of questions math olympiads ask, never lifted or reworded from any past contest, so you can hand it to your child without wondering where it came from.
MOEMS Division E Practice Problems (Grades 4–6)
Division E leans on solid arithmetic, fractions and an appetite for puzzles, not formal algebra. Try each problem on its own before checking the solution. If a strategy doesn’t click right away, that’s useful information, not a failure.
Problem 1. Nora spent half her birthday money on a book. Then she spent $4 on stickers. She has $6 left. How much birthday money did she start with?
Solution: Work backwards. Before the stickers she had $6 + $4 = $10, and that $10 was half of her money. So she started with $20.
Problem 2. On Monday, Malik read 1/4 of a 200-page book. On Tuesday he read 1/5 of what was left. How many pages has he read in total?
Solution: Monday: 200 × 1/4 = 50 pages, leaving 150. Tuesday: 150 × 1/5 = 30 pages. Total read: 50 + 30 = 80 pages.
Problem 3. Lockers in a hallway are numbered 1 to 50. A red sticker goes on every 3rd locker, a blue sticker on every 4th locker. How many lockers get both stickers?
Solution: A locker gets both only if its number divides evenly by 3 and by 4, which means it’s a multiple of 12. Between 1 and 50, that’s 12, 24, 36 and 48: 4 lockers.
Problem 4. Priya has $45. She buys 3 notebooks at $6 each and 2 pens at $2.50 each. How much does she have left?
Solution: Notebooks cost 3 × $6 = $18. Pens cost 2 × $2.50 = $5. Together that’s $23 spent, so $45 − $23 = $22 left.
Problem 5. A rectangular garden is 3 times as long as it is wide, and its perimeter is 64 feet. What is its area?
Solution: The perimeter is 2 widths plus 2 lengths, and since the length is 3 widths, that’s 8 widths total. 8 widths = 64 feet, so 1 width = 8 feet and the length is 24 feet. Area: 8 × 24 = 192 square feet.
MOEMS Division M Practice Problems (Grades 6–8)
Division M adds pre-algebra, ratios, number theory and problems that take more than one step to unravel. The jump from Division E is less about bigger numbers and more about setting up a relationship before you solve it.
Problem 1. At a school fair, the ratio of blue tickets sold to green tickets sold was 3:5. After 24 more green tickets were sold, the ratio became 3:8. How many blue tickets were sold?
Solution: Only the green count changed, so let blue = 3x and green = 5x. After the increase, 3x : (5x + 24) = 3 : 8, which gives 24x = 15x + 72, so x = 8. Blue tickets: 3 × 8 = 24.
Problem 2. What is the smallest positive integer that leaves a remainder of 4 when divided by 5, a remainder of 3 when divided by 4, and a remainder of 2 when divided by 3?
Solution: Each remainder is exactly 1 less than its divisor, so the number is 1 less than a common multiple of 3, 4 and 5. Their smallest common multiple is 60, so the answer is 60 − 1 = 59.
Problem 3. Three consecutive even integers add up to 96. What is the product of the smallest and the largest?
Solution: Call the integers n − 2, n and n + 2. Their sum is 3n = 96, so n = 32. The integers are 30, 32 and 34. Smallest times largest: 30 × 34 = 1,020.
Problem 4. Pipe A fills a tank in 12 minutes. Pipe B fills the same tank in 18 minutes. Working together, how many minutes will it take to fill the tank?
Solution: In one minute, Pipe A fills 1/12 of the tank and Pipe B fills 1/18. Together that’s 3/36 + 2/36 = 5/36 of the tank per minute. Filling the whole tank takes 36 ÷ 5 = 7 1/5 minutes.
Problem 5. A square and an equilateral triangle have the same perimeter. The triangle’s side is 8 cm longer than the square’s side. What is the area of the square?
Solution: Let the square’s side be s. The square’s perimeter is 4s and the triangle’s is 3(s + 8). Setting them equal: 4s = 3s + 24, so s = 24. Area: 24 × 24 = 576 square centimeters.
How to Use These Problems to Prepare
Don’t check the solutions right away. The real MOEMS format is five problems in 30 minutes, so try these the same way: pick five, set a 30-minute timer, and don’t look at a solution until time is up or your child is genuinely stuck. Missing a problem under real conditions teaches more than getting it right with unlimited time does. Our guide to how to solve math olympiad problems covers the actual strategies, drawing a diagram, working backwards, trying a smaller case, that unlock problems like these once the timing pressure is off.
A rotation that works for most families: one timed set a week, then spend the following session only on the misses, redoing them from scratch instead of just rereading the solution.
One detail is worth knowing before your child sits down: MOEMS gives one point for a right answer and takes nothing away for a wrong one, so guessing never actually costs anything. Gonit’s app runs on the same idea. Every question offers a hint before your child answers and a full explanation after a wrong one, so a miss turns into information instead of a losing move. It’s one option for the weeks between contests, not a replacement for a good teacher or a stack of papers your child actually sits down and works through.
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Bianca R. Samuels holds a Bachelor's degree in Mathematics and writes about building strong problem-solving foundations for young learners. Based in the United States, she focuses on making early math concepts clear, practical, and genuinely engaging for students and parents alike.
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