A self-study workbook of 100 rides through Counting, Probability, Number Theory, Algebra, and Geometry — built to prepare you for AIME-style competition math, one attraction at a time.
Solve the system 2x + 3y = 17 and x - y = 1. Find y.
Express x in terms of y from the second equation, then substitute into the first.
Systems of Equations (Substitution)
x = y + 1, so 2(y+1) + 3y = 17 gives 5y = 15, so y = 3.
Ticket #030GEO★★☆☆☆
Find the midpoint of the segment from (2,3) to (8,7).
Average the x-coordinates and average the y-coordinates.
Midpoint Formula
((2+8)/2, (3+7)/2) = (5,5).
Ticket #031COUNT★★☆☆☆
How many 3-person committees can be formed from 7 people?
Order doesn't matter, so use a combination, not a permutation.
Combinations
C(7,3) = 35.
Ticket #032PROB★★☆☆☆
A bag has 4 white and 6 black balls. Two balls are drawn WITH replacement. What is the probability both are white?
With replacement, each draw is independent with the same probability.
Independent Events with Replacement
(4/10) x (4/10) = 16/100, which simplifies to 4/25.
Ticket #033NUM★★☆☆☆
Find the least common multiple of 8 and 12.
Use the relationship LCM x GCD = product of the two numbers.
Least Common Multiple
GCD(8,12)=4, so LCM = (8x12)/4 = 24.
Ticket #034ALG★★☆☆☆
Two numbers have a sum of 15 and a difference of 3. Find the larger number.
Add the sum and the difference, then divide by 2 to get the larger number.
Sum and Difference System
(15+3)/2 = 9.
Ticket #035GEO★★☆☆☆
Find the slope of the line through (1,2) and (4,11).
Slope = (change in y) / (change in x).
Slope Formula
(11-2)/(4-1) = 9/3 = 3.
Ticket #036COUNT★★☆☆☆
In how many ways can 5 runners finish 1st, 2nd, and 3rd place (no ties)?
Order matters here, so use a permutation of 3 out of 5.
Permutations (nPr)
5x4x3 = 60.
Ticket #037PROB★★☆☆☆
What is the probability of flipping a fair coin 3 times and getting all tails?
Multiply (1/2) three times for three independent flips.
Independent Events
(1/2)^3 = 1/8.
Ticket #038NUM★★☆☆☆
Two positive integers have GCD 6 and LCM 72. If one of the integers is 18, what is the other?
Use the identity: GCD x LCM = product of the two numbers.
GCD-LCM Relationship
6 x 72 = 432 = 18 x (other number), so the other number = 432/18 = 24.
Ticket #039ALG★★☆☆☆
A set of three numbers has pairwise sums of 10, 12, and 14. Find the sum of all three numbers.
Adding all three pairwise sums counts each number exactly twice.
Pairwise Sum Systems
(10+12+14)/2 = 18.
Ticket #040GEO★★☆☆☆
A line has slope 2 and passes through (0,3). What is its y-value when x = 5?
Use the equation y = mx + b with the given slope and y-intercept.
Linear Equations of Lines
y = 2(5) + 3 = 13.
Ticket #041COUNT★★★☆☆
A pizza shop has 4 toppings. How many different pizzas have at least one topping?
Count all subsets of the toppings, then remove the empty topping set.
Subsets & Complement
2^4 - 1 = 15.
Ticket #042PROB★★★☆☆
A bag has 5 red and 3 blue marbles. Two are drawn WITHOUT replacement. What is the probability both are red?
After removing the first red marble, recompute the probability for the second draw from what remains.
Dependent Events
(5/8) x (4/7) = 20/56 = 5/14.
Ticket #043NUM★★★☆☆
Convert 25 (base 10) to base 2.
Repeatedly divide by 2 and record the remainders from bottom to top.
Base Conversion
25 = 16+8+1 = 11001 in base 2.
Ticket #044ALG★★★☆☆
Solve x^2 - 5x + 6 = 0. Find the larger root.
Factor the quadratic into two binomials.
Solving Quadratics
(x-2)(x-3)=0, so the roots are 2 and 3; the larger is 3.
Ticket #045GEO★★★☆☆
A triangle has vertices (0,0), (6,0), and (0,4). Find its area.
Since two sides lie on the axes, use them as base and height.
Area on the Coordinate Plane
(1/2) x 6 x 4 = 12.
Ticket #046COUNT★★★☆☆
From 5 men and 4 women, how many 3-person committees have exactly 2 men and 1 woman?
Choose the men and women separately, then multiply.
Combinations with Groups
C(5,2) x C(4,1) = 10 x 4 = 40.
Ticket #047PROB★★★☆☆
A die is rolled once. What is the probability of rolling an even number OR a number greater than 4?
Use P(A or B) = P(A) + P(B) - P(A and B), where A = even and B = greater than 4.
Union of Events
Even = {2,4,6}, greater than 4 = {5,6}, overlap = {6}. (3+2-1)/6 = 4/6 = 2/3.
Ticket #048NUM★★★☆☆
Convert 101101 (base 2) to base 10.
Multiply each binary digit by its place value (32,16,8,4,2,1) and add.
Base Conversion
32+0+8+4+0+1 = 45.
Ticket #049ALG★★★☆☆
Find the 10th term of the arithmetic sequence 3, 7, 11, 15, ...
Use a_n = a_1 + (n-1)d with common difference d = 4.
Arithmetic Sequences
3 + 4(9) = 3 + 36 = 39.
Ticket #050GEO★★★☆☆
Find the area of the triangle with vertices (1,1), (5,1), and (3,4).
Use the shoelace formula for triangle area from coordinates.
Shoelace Formula
Area = (1/2)|1(1-4)+5(4-1)+3(1-1)| = (1/2)|-3+15+0| = 6.
Ticket #051COUNT★★★☆☆
How many 2-element subsets of {A,B,C,D,E} do NOT include A?
Remove A first, then choose 2 from the 4 remaining letters.
Combinations with Restriction
C(4,2) = 6.
Ticket #052PROB★★★☆☆
A family has 2 children. Given that at least one is a girl, what is the probability both are girls? (Assume boy/girl are equally likely.)
List the equally likely outcomes {BB, BG, GB, GG}, then restrict to those with at least one girl.
Conditional Probability
Outcomes with at least one girl: BG, GB, GG (3 outcomes). Both girls: GG (1 outcome). Probability = 1/3.
Ticket #053NUM★★★☆☆
Convert 37 (base 10) to base 5.
Divide repeatedly by 5, tracking remainders.
Base Conversion
37 = 1(25) + 2(5) + 2(1), so 37 in base 5 is 122.
Ticket #054ALG★★★☆☆
Find the sum of the first 10 positive integers (1+2+...+10).
Use the formula n(n+1)/2.
Sum of an Arithmetic Series
10(11)/2 = 55.
Ticket #055GEO★★★☆☆
Two similar triangles have a similarity ratio of 2:3. If the area of the smaller triangle is 8, find the area of the larger triangle.
The ratio of areas equals the square of the ratio of corresponding sides.
Similar Triangles (Area Ratio)
Area ratio = (3/2)^2 = 9/4, so the larger area = 8 x 9/4 = 18.
Ticket #056COUNT★★★☆☆
How many 4-digit codes (digits 0-9, repeats allowed) start with an odd digit?
There are 5 odd choices for the first digit; the other 3 digits are unrestricted.
Multiplication Principle with Restriction
5 x 10 x 10 x 10 = 5000.
Ticket #057PROB★★★☆☆
A spinner has 4 equal sections numbered 1-4. It is spun twice. What is the probability the sum is 5?
List all pairs (a,b) with a+b=5 out of the 16 equally likely outcomes.
Sample Space Counting
Pairs summing to 5: (1,4),(2,3),(3,2),(4,1), so 4 out of 16 = 1/4.
Ticket #058NUM★★★☆☆
How many digits does 100 have when written in base 3?
Find the smallest power of 3 that exceeds 100; the exponent tells you the digit count.
Base Conversion (Digit Count)
3^4 = 81 and 3^5 = 243, and 81 <= 100 < 243, so 100 needs 5 digits in base 3.
Ticket #059ALG★★★☆☆
A geometric sequence has first term 2 and common ratio 3. Find the 5th term.
Use a_n = a_1 x r^(n-1).
Geometric Sequences
2 x 3^4 = 2 x 81 = 162.
Ticket #060GEO★★★☆☆
In similar triangles, corresponding sides measure 4 and 10. If the smaller triangle's perimeter is 12, find the larger triangle's perimeter.
The ratio of perimeters equals the ratio of corresponding sides.
Similar Triangles (Perimeter Ratio)
Ratio = 10/4 = 2.5, so the larger perimeter = 12 x 2.5 = 30.
Ticket #061COUNT★★★★☆
Find the number of subsets of {1,2,3,4,5} that are subsets of neither {1,2,3} nor {3,4,5}.
First find how many subsets of the 5-element set DO lie in at least one of the two smaller sets, using inclusion-exclusion, then subtract that from the total number of subsets.
Inclusion-Exclusion on Subsets
Subsets of {1,2,3}: 2^3=8. Subsets of {3,4,5}: 2^3=8. Subsets of both (subsets of {3}): 2^1=2. By inclusion-exclusion, subsets in at least one = 8+8-2 = 14. All subsets of the 5-set = 2^5 = 32. So subsets in neither = 32-14 = 18.
Ticket #062PROB★★★★☆
A box has 10 tickets numbered 1 through 10. One is drawn at random. What is the expected value of the number drawn?
Expected value equals the average of all equally likely outcomes.
Expected Value
(1+2+...+10)/10 = 55/10 = 5.5.
Ticket #063NUM★★★★☆
What is the remainder when 2024 is divided by 7?
Find the nearest multiple of 7 below 2024 and subtract.
Modular Arithmetic
7 x 289 = 2023, so 2024 mod 7 = 1.
Ticket #064ALG★★★★☆
Solve log_2(x) = 5. Find x.
Rewrite in exponential form: x = 2^5.
Logarithms
2^5 = 32.
Ticket #065GEO★★★★☆
A circle has area 49pi. Find its radius.
Set pi r^2 equal to the given area and solve for r.
Area of a Circle
r^2 = 49, so r = 7.
Ticket #066COUNT★★★★☆
How many ways can 6 people be seated around a circular table if rotations are considered identical?
Fix one person's seat to remove rotational symmetry, then arrange the rest.
Circular Permutations
(6-1)! = 5! = 120.
Ticket #067PROB★★★★☆
A game pays $10 if a fair coin lands heads and costs you $2 if it lands tails. What is the expected value of playing once?
Expected value = sum of (probability x outcome value) over all outcomes.
Expected Value
0.5(10) + 0.5(-2) = 5 - 1 = $4.
Ticket #068NUM★★★★☆
Find the last digit of 7^100.
The last digits of powers of 7 cycle with period 4: 7,9,3,1,7,9,3,1,...
Cyclicity of Last Digits
100 is a multiple of 4, so 7^100 ends in the same digit as 7^4, which is 1.
Ticket #069ALG★★★★☆
If log(x) = 2 (base 10), find x.
Rewrite in exponential form: x = 10^2.
Logarithms
10^2 = 100.
Ticket #070GEO★★★★☆
An equilateral triangle has side length 6. Find its area.
Use the formula (sqrt(3)/4) x side^2 for an equilateral triangle.
Equilateral Triangle Area
(sqrt(3)/4) x 36 = 9*sqrt(3).
Ticket #071COUNT★★★★☆
A coin is flipped 5 times. In how many outcomes are exactly 3 heads?
Choose which 3 of the 5 flips come up heads.
Combinations (Binary Outcomes)
C(5,3) = 10.
Ticket #072PROB★★★★☆
A bag has 6 red and 4 green balls. Two balls are drawn without replacement. What is the probability at least one is green?
Find the probability of NO green balls first, then subtract from 1.
Complementary Counting
P(no green) = (6/10)(5/9) = 30/90 = 1/3, so P(at least one green) = 1 - 1/3 = 2/3.
Ticket #073NUM★★★★☆
Find the remainder when 3^50 is divided by 5.
Powers of 3 mod 5 cycle with period 4: 3,4,2,1,3,4,2,1,...
Modular Arithmetic (Cyclicity)
50 mod 4 = 2, matching the 2nd term in the cycle (4), so the remainder is 4.
Ticket #074ALG★★★★☆
A sequence satisfies a_1 = 1 and a_(n+1) = a_n + n. Find a_5.
Compute each term one at a time using the recursive rule.
Recursive Sequences
a1=1, a2=1+1=2, a3=2+2=4, a4=4+3=7, a5=7+4=11.
Ticket #075GEO★★★★☆
A chord in a circle of radius 10 has length 12. Find the distance from the center to the chord.
Drop a perpendicular from the center to the chord's midpoint, forming a right triangle.
Chord-Radius Right Triangle
Half the chord is 6, so distance = sqrt(10^2-6^2) = sqrt(64) = 8.
Ticket #076COUNT★★★★☆
How many distinct ways can the letters of the word LEVEL be arranged?
Divide the total permutations of 5 letters by the factorial of each repeated letter's count.
Permutations with Repetition
LEVEL has 5 letters where L appears twice and E appears twice: 5!/(2!x2!) = 120/4 = 30.
Ticket #077PROB★★★★☆
Three fair coins are flipped. What is the probability of getting at least one head?
Use the complement of 'no heads at all'.
Complementary Counting
P(no heads) = (1/2)^3 = 1/8, so P(at least one head) = 1 - 1/8 = 7/8.
Ticket #078NUM★★★★☆
How many integers between 1 and 100 (inclusive) are divisible by 6 but NOT by 4?
Count multiples of 6, then subtract those that are also multiples of 4 (i.e., multiples of 12).
Modular Counting
Multiples of 6 up to 100: 16. Multiples of 12 (both 6 and 4): 8. So 16 - 8 = 8.
Ticket #079ALG★★★★☆
Solve 2^(x+1) = 32. Find x.
Write 32 as a power of 2, then match the exponents.
Exponential Equations
32 = 2^5, so x+1 = 5, giving x = 4.
Ticket #080GEO★★★★☆
A square has vertices (0,0), (6,0), (6,6), (0,6). Find the area of the region closer to (0,0) than to (6,6).
The perpendicular bisector of the two corners splits the square exactly in half.
Perpendicular Bisector Regions
The line x+y=6 splits the 36-area square into two equal halves of 18 each.
Ticket #081COUNT★★★★★
A club has 8 members. A president, secretary, and treasurer (three different people) must be chosen. In how many ways can this be done?
Assign 3 distinct roles out of 8 people; order matters since the roles differ.
Permutations (nPr)
8x7x6 = 336.
Ticket #082PROB★★★★★
Team A beats Team B in any single game with probability 2/3, independently each game. What is the probability A wins at least 2 of 3 games?
Add the probability of exactly 2 wins to the probability of exactly 3 wins, using the binomial formula.
How many positive integers less than 50 are relatively prime to 15?
Numbers relatively prime to 15 avoid multiples of 3 and of 5.
Euler's Totient Idea
Every block of 15 consecutive integers contains 8 numbers coprime to 15. Three full blocks (1-45) give 24, and checking 46,47,48,49 adds 3 more (46,47,49 qualify, 48 does not): 24+3 = 27.
Ticket #094ALG★★★★★
Find the sum of all values of n satisfying n^2 - 8n + 12 = 0.
Factor the quadratic or use the sum-of-roots formula -b/a.
Sum of Roots
(n-2)(n-6)=0, giving roots 2 and 6, which sum to 8.
Ticket #095GEO★★★★★
A 3x3x3 cube is built from 27 unit cubes. How many unit cubes touch the outer surface?
Subtract the single interior cube from the total.
3D Counting
Only the very center cube is fully interior, so 27 - 1 = 26 touch the surface.
Ticket #096COUNT★★★★★
Find the number of distinct ways to arrange the letters of BANANA.
Divide 6! by the factorial of the count of each repeated letter.
Permutations with Repetition
B appears once, A appears three times, N appears twice: 6!/(3!x2!) = 720/12 = 60.
Ticket #097PROB★★★★★
Two dice are rolled. Given that the sum is 8, what is the probability that at least one die shows a 6?
List all (a,b) pairs summing to 8, then count how many include a 6.
Conditional Probability
Sum = 8 pairs: (2,6),(3,5),(4,4),(5,3),(6,2), so 5 outcomes total. Those containing a 6: (2,6) and (6,2), so 2 outcomes. Probability = 2/5.
Ticket #098NUM★★★★★
How many trailing zeros does 20! have?
Count the factors of 5 in the product, since factors of 2 are more plentiful.
Trailing Zeros in Factorials
floor(20/5) + floor(20/25) = 4 + 0 = 4.
Ticket #099ALG★★★★★
A geometric sequence has first term 5 and 4th term 135. Find the common ratio.
Use a_4 = a_1 x r^3 and solve for r.
Geometric Sequences (Solving for r)
5r^3 = 135, so r^3 = 27, giving r = 3.
Ticket #100GEO★★★★★
In a 4x4 grid of points, how many distinct lines pass through exactly 4 of these points?
Count the horizontal, vertical, and diagonal lines that hit all 4 points in a row.