Let S be a set consisting of m pairs (a, b) of positive integers with the property that 1 ≤ a < b ≤ n. Show that there are at least

triples (a, b, c) such that (a, b), (a, c), and (b, c) belong to S.

[APMO 1989 Q 4]

Let A

[APMO 1989 Q 3]

The triangle ABC has lengths AB = 60, AC = 63 and BC = 39. The point X lies on the circumcircle of ABC such that the triangle AXB also has all integral sides.

Find the perimeters of all possible integral triangles AXB.

Note that from next week, the PMQ will be renamed Professor Pailyn's Mathematical Quiz - so still a PMQ!

Find the perimeters of all possible integral triangles AXB.

Note that from next week, the PMQ will be renamed Professor Pailyn's Mathematical Quiz - so still a PMQ!

Let ABC be a triangle whose angles α = CAB and β = CBA are greater than 45°.

Above the side AB we construct a right-angled isosceles triangle ABR with AB as hypotenuse, such that R lies*inside *the triangle ABC.

Analogously we erect above the sides BC and AC right-angled isosceles triangles CBP and ACQ, but with their (right-angled) vertices P and Q*outside *of the triangle ABC.

Show that CQRP is a parallelogram.

[OEMO Beginners 2001 Q4]

Above the side AB we construct a right-angled isosceles triangle ABR with AB as hypotenuse, such that R lies

Analogously we erect above the sides BC and AC right-angled isosceles triangles CBP and ACQ, but with their (right-angled) vertices P and Q

Show that CQRP is a parallelogram.

[OEMO Beginners 2001 Q4]

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

[OEMO Beginners 2001 Q1]

[OEMO Beginners 2001 Q1]

Let

[OEMO Beginners 2000 Q1]

Let n be a positive integer. Two players, Alice and Bob, are playing the following game:

• Alice chooses n real numbers, not necessarily distinct;

• Alice writes all pairwise sums on a sheet of paper and gives it to Bob (there are n(n-1)/2 such sums, not necessarily distinct);

• Bob wins if he finds correctly the initial n numbers chosen by Alice with only one guess.

Can Bob be sure to win for the following cases?

a) n = 5 b) n = 6 c) n = 8

Justify your answer(s).

[For example, when n = 4, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]

[Junior Balkan MO 2013 Problem 4][Note that the original paper has 4 problems to be done in 4.5 hours]

The figure ABCDEF is a cyclic hexagon with lengths AB = CD = EF = 5, BC = DE = 2 and AF = 11. Given that the area of the hexagon is 54, find the rational value of the length BE.

As usual, assume this is a competition question. You have an ample supply of paper and pens and... that's it! Of course, this

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]

When Tanvir climbed the Tajingdong mountain, on his way to the top he saw it was raining 11 times. At Tajindong, on a rainy day, it rains either in the morning or in the afternoon; but it never rains twice in the same day. On his way, Tanvir spent 16 mornings and 13 afternoons without rain. How many days did it take for Tanvir to climb the Tajindong mountain in total?

[Bangladesh BdMO 2012. This question appeared in the Primary, Junior and Secondary competitions.]

[Bangladesh BdMO 2012. This question appeared in the Primary, Junior and Secondary competitions.]

Let ABCDEFG be one half of a regular dodecagon.

Let P be the intersection of the lines AB and GF and let Q be the intersection of the lines AC and GE.

Show that Q is the circumcenter of the triangle AGP.

[OEMO Beginners 2000 Q4]

Let ABC be an acute triangle and D be a point on the altitude through C.

Prove that the mid-points of the line segments AD, BD, BC and AC form a rectangle.

[OEMO Beginners 2013 Q 4, by G. Anegg, Innsbruck]

Prove that the mid-points of the line segments AD, BD, BC and AC form a rectangle.

[OEMO Beginners 2013 Q 4, by G. Anegg, Innsbruck]

Imagine a circle with six distinct points marked on its circumference and labelled 1 to 6. There are also six cards numbered 1 to 6. The cards are shuffled and placed face down on a table. You pick two cards and look at the numbers on them; join the two points shown on the cards by a straight-line chord. You then pick two more cards and join those two points with a chord. Finally, a chord is drawn joining the last remaining pair of points.

You win the game if none of the chords intersect. What is the probability of winning this game?

Suppose there are 997 points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least 1991 red points in the plane.

Can you find a special case with exactly 1991 red points?

[APMO 1991 Q2]

Can you find a special case with exactly 1991 red points?

[APMO 1991 Q2]

Let G be the centroid of triangle ABC and M be the midpoint of BC. Let X be on AB and Y on AC such that the points X, Y and G are collinear and XY and BC are parallel.

Suppose that XC and GB intersect at Q and YB and GC intersect at P. Show that triangle MPQ is similar to triangle ABC.

[APMO 1991 Q1]

Suppose that XC and GB intersect at Q and YB and GC intersect at P. Show that triangle MPQ is similar to triangle ABC.

[APMO 1991 Q1]

Find all integers n > 1 such that the sum of n and its second-largest divisor is 2013.

[by R. Henner, Vienna. Austrian MO, Beginners' Competition 2013, Question 1]

[by R. Henner, Vienna. Austrian MO, Beginners' Competition 2013, Question 1]

Determine all finite non-empty sets S of positive integers satisfying

(i + j)/gcd(i, j) is an element of S for all i, j in S.

The gcd(i, j) is the greatest common divisor of i and j.

[APMO 2004 Q1][edited so easier to read online]

Yesterday, we looked at a problem involving a spice trader and his weights. Today,

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?

A spice trader was getting ready to go to market. He uses a balance (as in the diagram) and a set of weights measured in ounces. He wishes to weigh all whole number ounces from 1 to 32 inclusive.

He would also like to carry the minimum number of weights and the minimum sum of the weights.

Which weights should he take with him?

He would also like to carry the minimum number of weights and the minimum sum of the weights.

Which weights should he take with him?

During a break, n children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of n for which eventually, perhaps after many rounds, all children will have at least one candy each.

[APMO 1991 Q4]

Determine all pairs (a, b) of integers with the property that the numbers a

[APMO 1999 Q4]

Let a

(i) Any two of the a

(ii) Each a

Determine the smallest possible values of n to make sure that the sequence will contain a prime number.

[APMO 1995 Q2][with minor edit of typo]

I am posting this question as I think it is interesting, however, I also feel it needs some interpretation. I have copied it as written (apart from one typo correction), but I'm not sure why it asks for "values of n" in the plural. I assume the question is asking us to firstly find the maximum number of terms such that conditions (i) and (ii) are satisfied but without any primes appearing; then by adding one unused prime to such a sequence we would have found the minimum number that guarantees that a prime be present. I also assume that the integers are 2 to 1995 inclusive. I leave these assumptions open for discussion; it may be a case of lost in translation as APMO is the Asian Pacific MO.

During a break, a class of 7 children sit in a circle around their teacher to play a game. The teacher has a bag of 12 sweets and wishes to hand them all out. Knowing that some children will get more than others, he has devised this game to distribute them 'fairly'. He walks clockwise close to the children and hands out the sweets to some of them according to the following rule. He selects one child and gives him a sweet, then he skips the next child and gives a sweet to the next one, then he skips 2 and gives a sweet to the next one, then he skips 3, and so on.

After he has handed out all the sweets, the children look bemused - the game hasn't really worked very well! How many of the children end up with no sweets at all?

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