MATHCOUNTS Practice Problems: Original Questions for Every Round

MATHCOUNTS’s own website has a PDF called “2021 Handbook 200 Problems NO Solutions.” Two hundred real contest problems, filed under that exact name, with nothing showing how to get from problem to answer.

It’s not a one-off. Some years the official archive includes full worked solutions, some years just a bare answer key, and some years, like that one, nothing past the questions themselves.

If our MATHCOUNTS guide already got you through registration and the four rounds, you’re past that stage and you need something to solve.

These MATHCOUNTS practice problems are for exactly that: one original problem set for every real round, Sprint through Countdown, each one worked out in full.

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Why We Wrote Original MATHCOUNTS Practice Problems (Not Reposted Ones)

In the two days since the MATHCOUNTS guide went live, Google has already surfaced gonit.app for 12 distinct MATHCOUNTS-related searches.

Twenty-eight impressions total, average position around 2. Small numbers; this is a two-day-old page, but three of the twelve searches are the exact wording of one specific past contest problem: “at the mc theater mathcounts answer,” “mc theater 8:15 mathcounts,” and “at the mc theater mathcounts solution.” Notice what that student was actually looking for. Not a lesson on rates or ratios.

The answer to the one problem that beat them, word for word, because nothing in the official archive explained it.

That gap sits between two things that already exist.

The official archive is real and free, but inconsistent year to year. The next tier up, AoPS’s Alcumus and Trainer tools, is excellent but assumes you’re already fluent in that ecosystem, not looking for a first problem set to try cold. Nobody occupies the middle: original, beginner-facing, in the real round formats, explained all the way through instead of handed to you as a bare number.

Every problem below is ours. We wrote each one from scratch in the real MATHCOUNTS round format, and none of them reproduces or reworks an official contest question.

MATHCOUNTS owns its past problems as competition IP. If you’re weighing whether this is even the right contest for the season, see where MATHCOUNTS ranks among the major math competitions first. Its own free archive at mathcounts.org/resources/past-competitions is where the real past contests live, gaps and all.

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Sprint Round Practice (2 Original Problems, Fully Explained)

Sprint is 30 problems in 40 minutes, no calculator. Under a minute and a half a problem on average, which means the fastest path to the answer beats the cleverest one.

A number theory problem like the first one below usually hides a shortcut in plain sight. An algebra problem like the second one rewards setting up the equation fast and trusting it, not re-checking every line by hand.

Solution: The slow way is to list three-digit numbers with consecutive increasing digits (123, 234, 345, and so on) and check each one for divisibility by 3. At Sprint speed, don’t. Call the digits d, d+1 and d+2. The number itself equals 111d + 12, and its digit sum is 3d + 3, which factors to 3(d + 1). That’s divisible by 3 for every whole number d, meaning every number built this way already qualifies, not just some of them. Once you see that, the question stops being “which ones work” and becomes “which is smallest.” The smallest valid d is 1 (any smaller and the number doesn’t have three digits), giving digits 1, 2, 3. The answer is 123.

Solution: Two unknowns, one sentence relating them: that’s the signal to substitute, not to guess and check. Let the width equal w. The length is then 2w + 3, and the perimeter equation is 2(w + 2w + 3) = 54, which simplifies to 6w + 6 = 54, so w = 8. Swap that back into the length expression: 2(8) + 3 = 19. Multiply width by length: 8 x 19 = 152 square feet. Check it against the original perimeter if there’s time left: 2(8 + 19) = 54. It holds.

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Target Round Practice (2 Original Problems, Fully Explained)

Target is 8 problems worked in pairs, six minutes per pair, calculator allowed. Six minutes for two problems sounds generous until four of them go to the first one. Split the time on purpose, three minutes each, then move on and come back if there’s time left.

Solution: This is an inequality wearing a word-problem costume. Set it up directly: 0.85n + 12.50 is less than or equal to 60. Subtract 12.50 from both sides to get 0.85n is less than or equal to 47.50, then divide by 0.85 to get n is less than or equal to 55.88. The calculator hands you that decimal without blinking. The problem is what to do with it: n has to be a whole number of pencils, and rounding up blows the budget, so round down. The club can order 55 pencils, with $0.75 left over.

Solution: Averages hide a sum. Uncover it before touching the new number. Five tests averaging 88 means the five scores sum to 440 (5 x 88). Six tests averaging 90 means the six scores sum to 540 (6 x 90). The sixth score is whatever closes that gap: 540 minus 440 is 100. That’s a perfect score, and it’s the only number that raises a five-score average by two full points with just one more test. A calculator makes the multiplication instant. It won’t set up the subtraction for you.

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Team Round Practice (1 Original Problem, Fully Explained)

Team Round is 10 problems in 20 minutes, calculator allowed, and only the four students on the official team submit an answer. Real Team Round problems often chain through more than one idea at once, which is exactly what makes them a team event and not four students solving the same problem separately.

Solution: Work backward through each teammate in order, because each one depends on the last. A’s number: 3x = 51, so x = 17. B’s number: 17 + 17 = 34. C’s number: 34 x 2 = 68. D’s number: 68 minus 9 is 59. Swap any two teammates’ steps and the chain breaks, the same way a real Team Round problem falls apart if one student skips ahead without checking what the last student actually handed them.

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Countdown Round Practice (1 Fast Problem, Fully Explained)

Countdown gives you 45 seconds a problem, no calculator, head-to-head against another student on a buzzer. Whoever answers first, correctly, keeps going. There’s no time to write out long division or grind through repeated multiplication. You need a pattern, and you need to spot it fast.

Solution: Multiplying out 7^2026 by hand isn’t happening in 45 seconds, and it doesn’t need to. Powers of 7 cycle through their units digit every four steps: 7, 9, 3, 1, then back to 7 again. The units digit only depends on where the exponent lands in that four-step cycle, not on the full value of the power. Divide the exponent by 4: 2026 divided by 4 is 506 remainder 2. A remainder of 2 lands on the second number in the cycle. The units digit is 9. On the buzzer, you don’t compute 7^2026. You find the cycle and count.

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How to Use These Problems to Prepare

Six problems won’t get you through a season. Use them to check something more specific: can you get the right answer without help, and if not, exactly where does the reasoning break down?

That second question is where most free resources go quiet. MATHCOUNTS’s own archive frequently ships a bare answer key, a number with nothing behind it, so a wrong answer tells you that you’re wrong and nothing else. Gonit’s practice mode is built to do the opposite: a hint before you answer if you’re stuck, and a full step-by-step explanation after a wrong one, so you can see exactly which step went wrong instead of just how far off the number was. It’s one option among several ways to practice, not the only one, but it’s built to answer the specific gap these six problems can’t close by themselves.

Six problems is a starting point, not a full prep plan. Come back for another set once these are done, and pay attention to which round you’re actually weakest on. That’s the number worth optimizing, not just how many problems you finished. For the general strategy behind problems like these, read our guide to solving Olympiad-style problems; once there’s a MATHCOUNTS score to compare, what a good AMC score looks like explains more of that scoring logic than you’d expect.

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Related guides

More practice, one competition over. Two more guides pick up where these MATHCOUNTS problems leave off.

• AMC 8 Practice Test: How to Practice the Right Way
• Math Kangaroo vs AMC: Which Math Competition Should Your Child Start With?

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