81 real olympiad problems, with training wheels off

Genuine competition problems — IMO problems from 1959 to 1988 and the classics every olympiad student eventually meets — across algebra, combinatorics, geometry and number theory. Each takes 30–60 minutes and demands a proof. On Lemma every problem carries progressive hints, a full walkthrough and a marking rubric, so you can train the honest way: attempt, struggle, then mark yourself like a coordinator would.

Algebra (21)

• IMO 1964, Problem 2Let a, b, c be the sides of a triangle. Prove that a^2(b+c-a) + b^2(c+a-b) + c^2(a+b-c) ≤ 3abc. · 5/6 · ~45 min
• Functional equationFind all functions f : mathbbR to mathbbR satisfying f(x+y) = f(x) + f(y) for all real x,y, given that f is… · 4/6 · ~35 min
• AM–GMProve that for positive reals a,b,c: (a+b)(b+c)(c+a) ≥ 8abc. · 3/6 · ~30 min
• NesbittProve that for positive reals, (a)/(b+c) + (b)/(c+a) + (c)/(a+b) ≥ (3)/(2). · 4/6 · ~35 min
• Let a,b,c be reals with a+b+c = 0. Prove that a^3+b^3+c^3 = 3abc.Let a,b,c be reals with a+b+c = 0. Prove that a^3+b^3+c^3 = 3abc. · 4/6 · ~35 min
• Prove that if x + dfrac1x is an integer, then x^n + dfrac1x^n is an…Prove that if x + dfrac1x is an integer, then x^n + dfrac1x^n is an integer for every positive integer n. · 5/6 · ~40 min
• Let P(x) be a polynomial with integer coefficients. Prove that if…Let P(x) be a polynomial with integer coefficients. Prove that if P(a) = P(b) = P(c) = 1 for three distinct… · 5/6 · ~45 min
• Find all functions f:mathbbRtomathbbR such that f(x)f(y) - f(xy) = x…Find all functions f:mathbbRtomathbbR such that f(x)f(y) - f(xy) = x + y for all reals x,y. · 6/6 · ~50 min
• Prove that √(2) + √(3) is irrational.Prove that √(2) + √(3) is irrational. · 4/6 · ~35 min
• Prove that among any n+1 numbers chosen from \1,2,…,2n\, some one…Prove that among any n+1 numbers chosen from \1,2,…,2n\, some one divides another. · 5/6 · ~40 min
• Prove that for all reals x, x^4 - x + tfrac12 > 0.Prove that for all reals x, x^4 - x + tfrac12 > 0. · 4/6 · ~35 min
• Let a,b,c>0 with abc=1. Prove a+b+c ≥ tfrac1a+tfrac1b+tfrac1c is…Let a,b,c>0 with abc=1. Prove a+b+c ≥ tfrac1a+tfrac1b+tfrac1c is FALSE in general, and determine when… · 5/6 · ~40 min
• Prove that a polynomial of degree n with real coefficients has at…Prove that a polynomial of degree n with real coefficients has at most n real roots. · 5/6 · ~45 min
• Prove that (a^2+b^2)/(2) ≥ ((a+b)/(2))^2 for all reals, and determine…Prove that (a^2+b^2)/(2) ≥ ((a+b)/(2))^2 for all reals, and determine equality. · 4/6 · ~35 min
• Find all pairs of positive integers (x,y) with x^y = y^x and x ≠ y,…Find all pairs of positive integers (x,y) with x^y = y^x and x ≠ y, and prove there are no others. · 6/6 · ~50 min
• Prove that if a+b+c = 0 then (a^2+b^2+c^2)/(2) · (a^5+b^5+c^5)/(5) =…Prove that if a+b+c = 0 then (a^2+b^2+c^2)/(2) · (a^5+b^5+c^5)/(5) = (a^7+b^7+c^7)/(7). · 5/6 · ~45 min
• Prove that there is no polynomial P with integer coefficients such…Prove that there is no polynomial P with integer coefficients such that P(n) is prime for every positive… · 6/6 · ~50 min
• Prove that for every real number x, x^2+1 ≥ 2x, with equality if and…Prove that for every real number x, x^2+1 ≥ 2x, with equality if and only if x=1. · 1/6 · ~10 min
• AM–GMProve that for nonnegative reals a,b: (a+b)/(2) ≥ √(ab), with equality if and only if a=b. · 1/6 · ~10 min
• Prove that for all reals a,b: a^2+b^2 ≥ 2ab. Then use this to prove…Prove that for all reals a,b: a^2+b^2 ≥ 2ab. Then use this to prove that for all reals a,b,c: a^2+b^2+c^2 ≥… · 2/6 · ~20 min
• AM–GMLet a,b,c be positive reals with abc=1. Prove that a+b+c ≥ 3. · 3/6 · ~25 min

Combinatorics (21)

• Ramsey, R(3,3)=6Prove that among any six people, there are either three who all know each other, or three who are all mutual… · 4/6 · ~35 min
• Mutilated chessboardTwo opposite corners are removed from an 8times8 chessboard. Prove that the remaining 62 squares cannot be… · 3/6 · ~25 min
• Handshake lemmaProve that in any finite gathering, the number of people who have shaken hands an odd number of times is even. · 2/6 · ~20 min
• Erdős–SzekeresProve that any sequence of n^2+1 distinct real numbers contains a monotone subsequence of length n+1… · 5/6 · ~45 min
• Prove that in any group of n ≥ 2 people, at least two have shaken…Prove that in any group of n ≥ 2 people, at least two have shaken hands the same number of times (within the… · 3/6 · ~30 min
• A 5times5 grid has a token on each square. Every token must move to…A 5times5 grid has a token on each square. Every token must move to an orthogonally adjacent square. Prove… · 4/6 · ~35 min
• Prove that any set of 10 distinct two-digit numbers has two disjoint…Prove that any set of 10 distinct two-digit numbers has two disjoint non-empty subsets with the same sum. · 5/6 · ~40 min
• Prove that in any 2-colouring of the integers \1,2,…,9\, there exist…Prove that in any 2-colouring of the integers \1,2,…,9\, there exist three integers of the same colour… · 5/6 · ~45 min
• Prove that any tournament (a complete graph with every edge directed)…Prove that any tournament (a complete graph with every edge directed) has a Hamiltonian path — an ordering of… · 4/6 · ~35 min
• Prove that any 2-colouring of the edges of K_6 contains at least two…Prove that any 2-colouring of the edges of K_6 contains at least two monochromatic triangles. · 6/6 · ~50 min
• Prove that binomn0 + binomn1 + … + binomnn = 2^n by a counting…Prove that binomn0 + binomn1 + … + binomnn = 2^n by a counting argument, not by the binomial theorem. · 4/6 · ~35 min
• Prove that among any 5 points inside a unit square, two are within…Prove that among any 5 points inside a unit square, two are within distance (sqrt2)/(2). · 4/6 · ~35 min
• Prove that any n points in the plane, not all collinear, determine at…Prove that any n points in the plane, not all collinear, determine at least n distinct lines. · 5/6 · ~45 min
• Prove that if n+1 integers are chosen from \1,2,…,2n\, two of them…Prove that if n+1 integers are chosen from \1,2,…,2n\, two of them are coprime. · 6/6 · ~50 min
• There are 100 lockers, all closed. Student k toggles every k-th…There are 100 lockers, all closed. Student k toggles every k-th locker, for k=1,…,100. Prove exactly the… · 5/6 · ~45 min
• Prove that in any sequence of mn+1 distinct reals, there is an…Prove that in any sequence of mn+1 distinct reals, there is an increasing subsequence of length m+1 or a… · 4/6 · ~35 min
• Prove that the number of subsets of \1,…,n\ with an even number of…Prove that the number of subsets of \1,…,n\ with an even number of elements equals the number with an odd… · 5/6 · ~45 min
• Prove that a set with n elements has exactly 2^n subsets.Prove that a set with n elements has exactly 2^n subsets. · 1/6 · ~10 min
• PigeonholeProve that among any 13 people, two must share a birth month. · 1/6 · ~10 min
• Prove that binomnk = binomnn-k for all integers 0 ≤ k ≤ n, using a…Prove that binomnk = binomnn-k for all integers 0 ≤ k ≤ n, using a combinatorial (bijective) argument rather… · 2/6 · ~20 min
• PascalProve Pascal's identity, binomnk = binomn-1k-1 + binomn-1k, using a combinatorial argument. · 2/6 · ~20 min

Geometry (17)

• Prove that the angle inscribed in a semicircle is a right angle…Prove that the angle inscribed in a semicircle is a right angle (Thales' theorem). · 3/6 · ~30 min
• Prove that the perpendicular bisectors of the sides of a triangle are…Prove that the perpendicular bisectors of the sides of a triangle are concurrent. · 4/6 · ~35 min
• Let ABCD be a cyclic quadrilateral. Prove that angle A + angle C =…Let ABCD be a cyclic quadrilateral. Prove that angle A + angle C = 180^circ. · 5/6 · ~40 min
• Prove Ptolemy's inequality: for any four points A,B,C,D in the plane,…Prove Ptolemy's inequality: for any four points A,B,C,D in the plane, AC · BD ≤ AB· CD + BC · AD, with… · 5/6 · ~45 min
• Prove that the medians of a triangle are concurrent and divide each…Prove that the medians of a triangle are concurrent and divide each other in ratio 2:1. · 4/6 · ~35 min
• Prove that in any triangle, the orthocentre H, centroid G and…Prove that in any triangle, the orthocentre H, centroid G and circumcentre O are collinear with HG = 2 GO… · 6/6 · ~50 min
• A point P lies inside an equilateral triangle. Prove that the sum of…A point P lies inside an equilateral triangle. Prove that the sum of the distances from P to the three sides… · 5/6 · ~40 min
• Prove that the nine-point circle of a triangle passes through the…Prove that the nine-point circle of a triangle passes through the three side midpoints, the three feet of the… · 6/6 · ~50 min
• Prove that the sum of the exterior angles of any convex polygon is…Prove that the sum of the exterior angles of any convex polygon is 360^circ. · 4/6 · ~35 min
• In triangle ABC, prove that the angle bisector from A divides BC in…In triangle ABC, prove that the angle bisector from A divides BC in the ratio AB:AC. · 5/6 · ~40 min
• Prove that in any triangle, the longest side is opposite the largest…Prove that in any triangle, the longest side is opposite the largest angle. · 5/6 · ~40 min
• Prove that among all triangles with a given perimeter, the…Prove that among all triangles with a given perimeter, the equilateral has the largest area. · 6/6 · ~50 min
• Prove that the diagonals of a parallelogram bisect each other.Prove that the diagonals of a parallelogram bisect each other. · 5/6 · ~40 min
• Prove that the sum of the interior angles of a triangle is 180°.Prove that the sum of the interior angles of a triangle is 180°. · 1/6 · ~15 min
• Prove that the base angles of an isosceles triangle are equal.Prove that the base angles of an isosceles triangle are equal. · 1/6 · ~15 min
• Prove that the diagonals of a parallelogram bisect each other.Prove that the diagonals of a parallelogram bisect each other. · 2/6 · ~20 min
• CircumcenterProve that the perpendicular bisectors of the three sides of a triangle are concurrent (meet at a single… · 3/6 · ~25 min

Number Theory (22)

• IMO 1959, Problem 1Prove that the fraction (21n+4)/(14n+3) is irreducible for every natural number n. · 3/6 · ~30 min
• IMO 1988, Problem 6Let a and b be positive integers such that ab+1 divides a^2+b^2. Prove that (a^2+b^2)/(ab+1) is a perfect… · 6/6 · ~60 min
• EuclidProve that there are infinitely many prime numbers. · 2/6 · ~20 min
• IrrationalityProve that √(2) is irrational. · 2/6 · ~20 min
• Prove that n^5 - n is divisible by 30 for every integer n.Prove that n^5 - n is divisible by 30 for every integer n. · 3/6 · ~30 min
• Prove that there are infinitely many primes of the form 4k+3.Prove that there are infinitely many primes of the form 4k+3. · 4/6 · ~35 min
• Prove that if 2^n - 1 is prime, then n is prime.Prove that if 2^n - 1 is prime, then n is prime. · 4/6 · ~35 min
• Find all positive integers n such that n^2 + 1 is divisible by n + 1,…Find all positive integers n such that n^2 + 1 is divisible by n + 1, and prove your list is complete. · 5/6 · ~40 min
• Prove that gcd(2^m - 1,; 2^n - 1) = 2^gcd(m,n) - 1.Prove that gcd(2^m - 1,; 2^n - 1) = 2^gcd(m,n) - 1. · 5/6 · ~40 min
• Prove that for every positive integer n there exist n consecutive…Prove that for every positive integer n there exist n consecutive positive integers, none of which is prime. · 6/6 · ~50 min
• Prove that the sum of the reciprocals of the first n positive…Prove that the sum of the reciprocals of the first n positive integers, H_n = 1 + tfrac12 + … + tfrac1n, is… · 4/6 · ~35 min
• Prove that if p is a prime greater than 3, then p^2 - 1 is divisible…Prove that if p is a prime greater than 3, then p^2 - 1 is divisible by 24. · 4/6 · ~35 min
• Prove that x^2 + y^2 = 3z^2 has no solutions in positive integers.Prove that x^2 + y^2 = 3z^2 has no solutions in positive integers. · 5/6 · ~40 min
• Prove that the equation a^2 + b^2 = c^2 with a,b,c positive integers…Prove that the equation a^2 + b^2 = c^2 with a,b,c positive integers forces at least one of a,b to be… · 5/6 · ~40 min
• Prove that there are infinitely many n for which n^2+1 has a prime…Prove that there are infinitely many n for which n^2+1 has a prime factor greater than 2n. · 6/6 · ~50 min
• Prove that the product of any four consecutive integers is divisible…Prove that the product of any four consecutive integers is divisible by 24. · 4/6 · ~35 min
• Prove that √(p) is irrational for every prime p.Prove that √(p) is irrational for every prime p. · 5/6 · ~40 min
• Prove that for any integer n>1, n^4 + 4 is composite.Prove that for any integer n>1, n^4 + 4 is composite. · 6/6 · ~50 min
• Prove that the sum of any two consecutive integers is odd.Prove that the sum of any two consecutive integers is odd. · 1/6 · ~10 min
• Prove that the square of any odd integer is odd.Prove that the square of any odd integer is odd. · 1/6 · ~10 min
• Prove that a positive integer N is divisible by 9 if and only if the…Prove that a positive integer N is divisible by 9 if and only if the sum of its digits is divisible by 9. · 2/6 · ~20 min
• Prove that consecutive integers are always coprime: gcd(n, n+1) = 1…Prove that consecutive integers are always coprime: gcd(n, n+1) = 1 for every integer n. · 2/6 · ~15 min