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Full AMC 10 Mock · No. 1

AMC 10 · 25 problems · 75 min
Full contest simulation: 25 problems in 75 minutes. Difficulty climbs sharply after #15. Year-8 students typically finish 10-15 problems on attempt #1 and run out of time on the rest — that's expected. AMC scoring: +6 correct, 0 wrong, +1.5 blank (random guessing across 5 options averages 1.2 points — less than blank). AMC 10 cutoff for AIME (top ~2.5%) is roughly 100 points (17 correct + 8 blank). A Year-8 strong attempt: 50-70 pts. Strategy: race through #1-#10 in ~20 min, spend 35 min on #11-#20, leave 20 min for #21-#25 and blank what you can't crack.
Year-8 notation glossary — open this BEFORE you start
(nk)\binom{n}{k} (binomial coefficient)
Number of ways to pick kk items from nn when order doesn't matter. (52)=10\binom{5}{2} = 10.
n!n! (factorial)
n!=n×(n−1)×(n−2)×⋯×1n! = n \times (n-1) \times (n-2) \times \dots \times 1. So 5!=1205! = 120.
a≡b(modm)a \equiv b \pmod{m}
aa and bb leave the same remainder when divided by mm. 13≡3(mod5)13 \equiv 3 \pmod 5 because 13=2×5+313 = 2 \times 5 + 3.
a mod ma \bmod m
The remainder when aa is divided by mm. 7 mod 5=27 \bmod 5 = 2.
Roots of a polynomial
The numbers xx that make the polynomial equal zero. Roots of x2−7x+12x^2 - 7x + 12 are 33 and 44.
⌊x⌋\lfloor x \rfloor (floor)
Round xx DOWN to an integer. ⌊5.7⌋=5\lfloor 5.7 \rfloor = 5, ⌊−3.2⌋=−4\lfloor -3.2 \rfloor = -4.
log⁡bx\log_b x
The power you raise bb to in order to get xx. log⁡28=3\log_2 8 = 3 because 23=82^3 = 8.
ii (imaginary unit)
A number with i2=−1i^2 = -1. A 'complex number' is a+bia + bi, like 3−4i3 - 4i.
∣3−4i∣|3 - 4i| (complex modulus)
Distance from zero in the plane: a2+b2\sqrt{a^2 + b^2}. So ∣3−4i∣=9+16=5|3 - 4i| = \sqrt{9 + 16} = 5.
How scoring works
  • +6 points for each correct answer.
  • 0 points for each wrong answer.
  • +1.5 points for each question left blank — so guessing randomly is usually worse than leaving it blank.
  • Max possible: 150 points.