INTERNATIONAL OLYMPIADS

INDIAN NATIONAL MATH OLYMPIAD 2014 (INMO 2014)

In a triangle ABC

, let

be the point on the segment

such that

. Suppose that the points

and the centroids of triangles ABD

and

lie on a circle. Prove that AB=AC

Let

be a natural number. Prove that,
$\left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfl...$
is even.

Let

be natural numbers with ab>2

. Suppose that the sum of their greatest common divisor and least common multiple is divisble by a+b

. Prove that the quotient is at most $\frac{a+b}{4}$ . When is this quotient exactly equal to $\frac{a+b}{4}$

Written on a blackboard is the polynomial x^2+x+2014

. Calvin and hobbes take turns alternatively(starting with Calvin) in the following game. During his turns alternatively(starting with Calvin) in the following game. During his turn, Calvin should either increase or decrese the coeffecient of

. And during this turn, Hobbes should either increase or decrease the constant coefficient by

. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

In a acute-angled triangle ABC

, a point

lies on the segment

. Let

denote the circumcentres of triangles ABD

and

respectively. Prove that the line joining the circumcentre of triangle ABC

and the orthocentre of triangle O_1O_2D

is parallel to

Let

be a natural number. Let $U=\{1,2,...,n\}$ , and define $A\Delta B$ to be the set of all those elements of

which belong to exactly one of

and

. Show that $|\mathcal{F}|\le 2^{n-1}$ , where $\mathcal{F}$ is a collection of subsets of

such that for any two distinct elements of A,B

of $\mathcal{F}$ we have $|A\Delta B|\ge 2$ . Also find all such collections $\mathcal{F}$ for which the maximum is attained.

SINGAPORE MATH OLYMPIAD (SENIOR) 2013

A shop sells two kind of products A and B. One day a salesman sold both A and B at the same price, $2100 to a customer. Suppose A makes a profit of 20% and B makes a loss of 20%. Then the deal
(A) make a profit of $70; (B) make a loss of $70;(C) make a profit of $175; (D) make a loss of $70;
Let f and g be functions such that for all real numbers x and y g(f(x+y)) = f(x) + (x+y)g(y).. Find the value of g(0) + g(1) + … + g(2013)
Each chocolate costs 1 dollar, each licorice stick costs 50 cents and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars.
Let A= {1, 2, 3, 4, 5, 6}. Find the number of distinct functions f: A → A such that f(f(f(n))) = n for all n ∈ A.
Find the number of triangles whose sides are formed by the sides of the diagonals of a regular heptagon (7 sided polygon). (Note: the vertices of the triangle need not be vertices of the heptagon).
Six seats are arranged in a circular table. Each seat is to be painted in red, blue or green such that any two adjacent seats have different colors. How many ways are there to paint the seats?
ΔABC is an equilateral triangle with side length 30. Fold the triangle so that A touches a point X on BC. If BX = 6, find the value of k, where √k is the length of the crease obtained from folding.
As shown in the figure below, circles C1 and C2 of radius 360 are tangent to each other, and both tangent to the straight line l. If the circle C3 is tangent to C1 , C2 and l, and circle C4 is tangent to C1 , C3 and l, find the radius of C4 .
Set {x} = x – ⌊x⌋, where ⌊x⌋ denotes the largest integer less than or equal to x. Find the number of real solutions to the equation {x} + {x2 } = 1 , |x| ≤ 10.
Find the value of ⌊$ ( \frac {3 + \sqrt 17 } {2} )^6 $⌋.
A regular dodecagon (12 sided polygon) is inscribed in a circle of radius 10. Find it’s area.

USAJMO 2012 QUESTIONS

Given a triangle ABC, let P and Q be the points on the segments AB and AC, respectively such that AP = AQ. Let S and R be distinct points on segment BC such that S lies between B and R, ∠BPS = ∠PRS, and ∠CQR = ∠QSR. Prove that P, Q, R and S are concyclic (in other words these four points lie on a circle).
Find all integers $n \ge 3 $ such that among any n positive real numbers $ a_1 , a_2 , … , a_n $ with
max $(a_1 , a_2 , … , a_n) \le n$ min $( a_1 , a_2 , … , a_n)$ there exist three that are the side lengths of an acute triangle.
Let a, b, c be positive real numbers. Prove that $\frac{a^3 + 3 b^3}{5a + b} + \frac{b^3 + 3c^3}{5b +c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2)$.
Let $\alpha$ be an irrational number with $0 < \alpha < 1$, and draw a circle in the plane whose circumference has length 1. Given any integer $n \ge 3 $, define a sequence of points $P_1 , P_2 , … , P_n $ as follows. First select any point $P_1$ on the circle, and for $ 2 \le k \le n $ define $P_k$ as the point on the circle for which the length of the arc $P_{k-1} P_k$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k-1} $ to $P_k$. Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a+b \le n$.
For distinct positive integers a, b < 2012, define f(a, b) to be the number of integers k with $1\le k < 2012$ such that the remainder when ak divided by 2012 is greater than that of bk divided by 2012. Let S be the minimum value of f(a, b), where a and b range over all pairs of distinct positive integers less than 2012. Determine S.
Let P be a point in the plane of triangle ABC, and $\gamma$ be a line passing through P. Let A’, B’, C’ be the points where reflections of the lines PA, PB, PC with respect to $\gamma$ intersect lines BC, AC, AB, respectively. Prove that A’, B’ and C’ are collinear.

VIETNAM NATIONAL MATH OLYMPIAD

Day 1 – 03 January 2014

1	Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and $\begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases}$ for all $n=1,2,3,\ldots$ . Prove that they are converges and find their limits.

2	Given the polynomial $P(x)=(x^2-7x+6)^{2n}+13$ where is a positive integer. Prove that can’t be written as a product of non-constant polynomials with integer coefficients.

3	Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let be the number of pairs of adjacent red vertices and be the number of pairs of adjacent blue vertices. a) Find all possible values of pair b) Determine the number of pairwise non-similar colorings of the polygon satisfying 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.

4	Let be an acute triangle, be the circumcircle, and Let be the midpoint of arc (not containing ). lies on $K\ne C$ such that intersects at the second point $D\ne B$ and intersects at intersects at a) Prove that $EF=\frac{BC}{2}.$ b) lies on such that is parallel to intersects at The circumcircle of triangle intersects at the second point Prove that passes through the midpoint of segment

Day 2 – 04 January 2014

1	Given a circle and two fixed points on and an arbitrary point on such that the triangle is acute. lies on ray lies on ray such that and Let be the intersection of and $P\ne A.$ intersects at a) Prove that are collinear. b) is the midpoint of Let be the intersection of and $K\ne A.$ is the line passing through and perpendicular to is the intersection of and intersects at $F\ne A.$ Prove that passes through a fixed point.

2	Find the maximum of $P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$ where are positive real numbers.

3	Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.