Sliced Parallelogram: OEMO 2003 Advanced Q4: Upper Secondary Mathematics Competition Question

In a parallelogram ABCD, let E be the midpoint of the side AB and F the midpoint of BC. Let P be the intersection point of the lines EC and F D.

Show that the segments AP, BP, CP and DP divide the parallelogram into four triangles with areas in 1 : 2 : 3 : 4 ratio.


[OEMO 2003 Federal Advanced Q4]

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]

OEMO 2001 Advanced Q1: Upper Secondary Mathematics Competition Question

Let n be an integer and S(n) be the sum of the 2001 powers of n with exponents 0 through 2000.
That is,


Determine the final digit (i.e., the ones-digit) in the decimal expansion of S(n).


[OEMO 2001 Advanced Q1]

OEMO 2001 Advanced Q4: Upper Secondary Mathematics Competition Question

Let A₀ = {1, 2} and for n > 0 let A_n be the set of all numbers that are either elements of A_n-1 or can be represented as the sum of two distinct elements of A_n-1.

Further let a_n = |A_n| be the number of elements of A_n.

Determine a_n as a function of n.

[OEMO 2001 Advanced Q4]

Peculiar Pentagon: Upper Secondary Mathematics Competition Question

In a convex pentagon ABCDE the areas of the triangles ABC, ABD, ACD and ADE are all equal to the same value F.

What is the area of the triangle BCE?


[OEMO 2001 Advanced Q3]

OEMO Beginners 2001 Q4: Middle Secondary Mathematics Competition Question

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]

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 Q2: Middle Secondary Mathematics Competition Question

Let a and b be positive real numbers. Prove that the inequality



holds.

When does equality hold?

[OEMO Beginners 2000 Q2]

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]

OEMO Beginners 2000 Q4: Middle Secondary Mathematics Competition Question

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]

OEMO Beginners 2013 Q3: Middle Secondary Mathematics Competition Questions

Let a and b be real numbers with 0 ≤ a, b ≤ 1.
Prove that



and find the cases of equality.


[OEMO Beginners 2013 Q 3, by K. Czakler, Vienna]

OEMO Beginners 2013 Q4: Middle Secondary Mathematics Competition Question

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]

OEMO Beginners 2013 Q1: Middle Secondary Mathematics Competition Question

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]