← All contests
Philippine Contest

PMO

59 problems from PMO, each rated for difficulty. Solve one to open its discussion.

The Philippine Mathematical Olympiad (PMO) is the oldest and most prominent nationwide mathematics competition among high school students. It is the first step for those who want to represent the country in the International Mathematical Olympiad (IMO).

It runs in three stages — an area qualifying stage, an area finals stage, and a national finals.

Problems

59 problems
A Right Angle Inside a Square Geometry 26th PMO Qualifying Stage 1382 14
Water Level in a Cylindrical Container Geometry 28th PMO Qualifying Stage 1333 9
Ratio from a Perfect Square's Digit Groups Number Theory 25th PMO Qualifying Stage 1483 11
Two Quadratic Equations with Four Unknowns Algebra 28th PMO Qualifying Stage 1608 16
Square and Rectangle in a Semicircle Geometry 26th PMO Qualifying Stage 1247 27
Length from an Orthocenter Configuration Geometry 25th PMO Qualifying Stage 1667 6
Numbers Divisible by 13 with a Digit-Sum Condition Number Theory 28th PMO Qualifying Stage 1527 30
Maximum Face Sum in a Minimal Divisor Cube Labeling Number Theory 28th PMO Qualifying Stage 1665 13
Sum of Radii of a Tangent Circle Geometry 28th PMO Qualifying Stage 1233 41
Value of a Recursively Defined Base-3 Function Number Theory 25th PMO Qualifying Stage 1570 22
Euler Totient x Geometric Series Algebra 21st PMO National Finals 1660 19
An Expression in the Roots Algebra 26th PMO Qualifying Stage 1173 50
Minimum Path Length in a Rectangle Geometry 28th PMO Qualifying Stage 1533 16
Probability of a Positive Score in a Coin Game Combinatorics 28th PMO Qualifying Stage 1480 18
Expressing a Variable from Equal Logarithms Algebra 28th PMO Qualifying Stage 1127 21
Largest Index Before a Sequence Becomes Non-Integer Number Theory 25th PMO Qualifying Stage 1583 12
Non-Attacking Coins on a Chessboard Combinatorics 27th PMO Qualifying Stage 1250 8
Shirts in a Drawer Combinatorics 27th PMO Qualifying Stage 1333 14
Digits from a Lexicographic Permutation Sequence Combinatorics 28th PMO Qualifying Stage 1583 9
Slopes of an Isosceles Triangle Geometry 26th PMO Qualifying Stage 1458 13
Stopping on the Third Draw Combinatorics 26th PMO Qualifying Stage 1177 20
Guessing a Two-Digit Number Logic 26th PMO Qualifying Stage 1435 37
Intersection Point of Two Logarithmic Graphs Algebra 25th PMO Qualifying Stage 1333 10
Length in a Perpendicular Bisector Configuration Geometry 25th PMO Qualifying Stage 1708 7
Ones in Binary from 1 to 128 Number Theory 27th PMO Qualifying Stage 1542 11
Largest Perfect Square Not Expressible as a Floor Sum Number Theory 25th PMO Qualifying Stage 1708 8
Minimizing the Largest Term of an Increasing Sequence Number Theory 25th PMO Qualifying Stage 1167 7
Painting a Prism Twice Geometry 27th PMO Qualifying Stage 1333 6
Point Inside a Triangle Combinatorics 28th PMO Qualifying Stage 1542 7
A Cubic with Symmetric Clues Algebra 27th PMO Qualifying Stage 1667 9
Area of a Rhombus Given Diagonals Geometry 26th PMO Qualifying Stage 963 24
The Goofy Number Problem Logic 27th PMO Qualifying Stage 1292 16
The Largest Prime Factor in a Factorial Sum Number Theory 26th PMO Qualifying Stage 1487 16
A Complex Root and Its Expression Algebra 27th PMO Qualifying Stage 1373 25
Smallest Circle Through Three Points Geometry 27th PMO Qualifying Stage 1247 21
Counting Terms in a Symmetric Expansion Combinatorics 18th PMO Qualifying Stage 1580 18
Twenty-Seven from Negative Seven Number Theory 27th PMO Qualifying Stage 1252 23
Minimizing a Product of Two Expressions Algebra 25th PMO Qualifying Stage 1435 28
Heads, Tails, and a Row of Coins Number Theory 26th PMO Qualifying Stage 1173 28
Diagonal of a Quadrilateral Geometry 25th PMO Qualifying Stage 1295 33
Dilating a Lattice Polygon Number Theory 26th PMO Qualifying Stage 1393 23
Avoiding Multiples of Twenty-Seven Number Theory 27th PMO Qualifying Stage 1468 34
A Pair of Mutually Dividing Primes Number Theory 26th PMO Qualifying Stage 1710 15
Average Pairwise Difference in a Set Combinatorics 25th PMO Qualifying Stage 1490 27
Coins in a Line Number Theory 28th PMO Qualifying Stage 1128 63
Counting Quadruples Satisfying a Sum Constraint Combinatorics 28th PMO Qualifying Stage 1590 18
Floors, Fractions, and a Quadratic Twist Algebra 27th PMO Qualifying Stage 1660 19
Parity of Power-Sum Divisors Number Theory 27th PMO Qualifying Stage 1678 17
Segment Length from Parallel Chords in a Circle Geometry 28th PMO Qualifying Stage 1320 18
Angle in a Regular Heptagon with a Length Condition Geometry 28th PMO Qualifying Stage 1542 13
The Angle Chain Quadrilateral Geometry 27th PMO Qualifying Stage 1160 20
A Recursive Angle Chase Geometry 27th PMO Qualifying Stage 1667 11
Counting Values of a Sum of Floor Functions Number Theory 28th PMO Qualifying Stage 1658 15
A Snake on a 3x3 Grid Combinatorics 27th PMO Qualifying Stage 1680 15
Angle in an Extremal Triangle Configuration Geometry 10th PMO 1250 7
Chord Length Between Tangency Points Geometry 10th PMO 1417 9
Counting Subsets with Divisibility Constraint Combinatorics PMO 2018 1500 11
Remainder of a Binomial Coefficient Sum Number Theory 26th PMO 1333 9
Symmetric Function of Quadratic Roots Algebra 26th PMO 1193 21