IrMO 2000 P1 Q5: Upper Secondary Mathematics Competition Question

Consider all parabolas of the form y = x² + 2px + q (p, q real) which intersect the x- and y-axes in three distinct points. For such a pair p, q let C(p,q) be the circle through the points of intersection of the parabola y = x² +2px+q with the axes. Prove that all the circles C(p,q) have a point in common.


[IrMO 2000, Paper 1, Question 5]

IrMO 1999 P2 Q1: Upper Secondary Mathematics Competition Question

Solve the system of (simultaneous) equations

y² = (x + 8)(x² + 2);

y² = (8 + 4x)y + 5x² - 16x - 16:


[IrMO 1999, Paper 2, Question 1]

IrMO 1999 P1 Q5: Upper Secondary Mathematics Competition Question

Three real numbers a, b, c with a < b < c, are said to be in arithmetic progression if c - b = b - a.

Define a sequence u(n), n = 0, 1, 2, 3, ... as follows: u(0) = 0, u(1) = 1 and, for each n > 0, u(n+1) is the smallest positive integer such that u(n+1) > u(n) and {u(0), u(1),... u(n), u(n+1)} contains no three elements that are in arithmetic progression.

Find u(100).


[Irish MO, Paper 1, Question 5, 1999]

Sums of Naturals: Middle Secondary Mathematics Competition Question

Let N be a natural number with the property that it is the sum of 3 consecutive natural numbers, of 4 consecutive naturals and also of 5 consecutive naturals.

Find the smallest value of N such that the 3 sequences above are disjoint, that is, there is no number that is in more than one sequence.

Four Perimeters: Upper Primary Mathematics Competition Question

A rectangle is divided into four smaller rectangles using two perpendicular lines, as shown in the diagram.

The number inside each small rectangle indicates the length of its perimeter. The diagram is not drawn to scale.

What value should go into the empty rectangle?

OEMO 2013 Advanced Q2: Upper Secondary Mathematics Competition Question

Determine all integers x satisfying



(Hint: [y] is the largest integer which is not larger than y.)


[OEMO 2013 Advanced Q2]

Area of Isosceles Triangle: Middle Secondary Mathematics Competition Question

Consider an isosceles triangle with sides (m, m, 2m-2). If the area of the triangle is 360, find the value of m.

OEMO Beginners 2001 Q2: Middle Secondary Mathematics Competition Question

We consider the quadratic equation x² - 2mx - 1 = 0, where m is an arbitrary real number.

For which values of m does the equation have two real solutions, such that the sum of their cubes equals eight times their sum.


[OEMO Beginners 2001 Q2]

OEMO Beginners 2001 Q1: Middle Secondary Mathematics Competition Question

Prove that for all odd positive integers n the number nⁿ-n is divisible by 24.



[OEMO Beginners 2001 Q1]

OEMO Beginners 2000 Q1: Middle Secondary Mathematics Competition Question

Let a be a real number. Determine for all a all pairs (x, y) of real numbers such that
(x - y²)(y - x²) + x³ + y³ = a holds.


[OEMO Beginners 2000 Q1]

OEMO Beginners 2000 Q3: Middle Secondary Mathematics Competition Question

A "nice" two-digit number is at the same time a multiple of the product of its digits and a multiple of the sum of its digits.

How many such two-digit numbers exist?

What is the quotient of number and sum of digits for each of these numbers?

[OEMO Beginners 2000 Q3]

OEMO Advanced 2000 Q3: Upper Secondary Mathematics Competition Question

Determine all real solutions of the equation




[OEMO Advanced 2000 Q3]

Prize Maths Quiz: A Four-Pan Balance Puzzle (PMQ35)

Yesterday, we looked at a problem involving a spice trader and his weights. Today, you are going to be the spice trader.

Imagine you have a four-pan balance, as illustrated below. The two outer pans are twice the distance from the fulcrum as the inner pans, the whole arrangement being balanced at the start.

You have a set of weights calibrated to be whole number ounces but you really don't want to carry them all with you. You wish to be able to weigh every amount between 0.5 to 32 ounces inclusive, going up in steps of 0.5 ounces, and to do so in one weighing.

What set of weights should you take, given that you want the smallest number of weights and the smallest sum of their weights?

Is your answer unique, or is there more than one solution?

Partitioning a Circle: Upper Secondary Mathematics Competition Question

This is the last of this week’s excursions into pizzas and cakes. In essence, these questions have been about the partitioning of two- and three-dimensional figures, so let’s abandon the culinary analogies for this one.

Take a circle and distribute n distinct points around its circumference. Join each point to every other point with a chord. The circle has thus been partitioned into a number of non-overlapping regions. Let R(n) be the maximum number of regions that can be created.

Given that R(n) is a quartic polynomial, or otherwise, find R(n) and hence calculate R(10).

Slicing Jupiter: Middle Secondary Mathematics Competition Question

What is the maximum number of pieces of cake that can be made with 10 planar cuts?

Let’s take a spherical cake, somewhat like the Jupiter layer cake in the image, and each planar knife cut slices the cake into two pieces. We’ve met this problem before, but beyond 4 or 5 slices trying to think geometrically can be a brain ache - unless you are gifted in spatial reasoning! However, the general formula for the maximum number of pieces N(p) after p cuts is just a cubic polynomial. Find this general formula and solve the problem!


[Note that the Jupiter Cake is for real and can be found at Cakecrumbs.]

[Note 2. The slice taken in the diagram, showing the inner layers, is not a planar cut.]

Nordic MC 1993 Q1: Upper Secondary Mathematics Competition Question

Let F be an increasing real function defined for all x, 0 ≤ x ≤ 1, satisfying the conditions

(i) F(x/3) = F(x)/2,

(ii) F(1 − x) = 1 − F(x).

Determine F(173/1993) and F(1/13).


[Nordic MC 1993 Q 1][slightly edited]

Nordic MC 1988 Q2: Upper Secondary Mathematics Competition Question

Let a, b, and c be non-zero real numbers and let a ≥ b ≥ c. Prove the inequality

When does equality hold?

[Nordic MC 1988 Q2]

Nordic MC 1994 Q4: Middle Secondary Mathematics Competition Question

Determine all positive integers n < 200, such that n² + (n + 1)² is the square of an integer.


[Nordic Mathematical Contest 1994 Q4]

IrMO 1990 P1 Q6: Upper Secondary Mathematics Competition Question

We recently had a question where considering the parity of the numbers (odd and evenness) speeded up our proof. Here is a question that is explicitly about the parity of numbers.


Let n be a natural number, and suppose that the equation

x₁x₂ + x₂x₃ + x₃x₄ + . . . + x_n-1x_n + x_nx₁ = 0

has a solution with each of the x_i's equal to either +1 or -1. Prove that n is divisible by 4.


[IrMO 1990 Paper 1 Question 6][edited slightly]

Irish MO 1988 P1 Q4: Upper Secondary Mathematics Competition Question

A mathematical moron is given the values b, c, A for a triangle ABC and is required to find the value of a. He does this by using the cosine rule

a² = b² + c² - 2bc.cosA

and misapplying the law of logarithms to this to get

log(a²) = log(b²) + log(c²) - log(2bc.cosA).

He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms and gets the correct answer. What can be said about the triangle ABC?

[ Irish MO 1988 Paper 1 Question 4]

The wording is genuine :-)