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IMO 2018 Problems – Day 2

July 10, 2018 Leave a comment

Problem 4. A site is any point {(x, y)} in the plane such that {x} and {y} are both positive integers less than or equal to 20.

Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to {\sqrt{5}}. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.

Find the greatest {K} such that Amy can ensure that she places at least {K} red stones, no matter how Ben places his blue stones.

Problem 5. Let {a_1,a_2,\ldots} be an infinite sequence of positive integers. Suppose that there is an integer {N > 1} such that, for each {n \geq N}, the number

\displaystyle \frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}

is an integer. Prove that there is a positive integer {M} such that {a_m = a_{m+1}} for all {m \geq M}.

Problem 6. A convex quadrilateral {ABCD} satisfies {AB\cdot CD = BC\cdot DA}. Point {X} lies inside {ABCD} so that {\angle{XAB} = \angle{XCD}} and {\angle{XBC} = \angle{XDA}}. Prove that {\angle{BXA} + \angle{DXC} = 180}.

Source: AoPS

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IMO 2018 Problems – Day 1

July 9, 2018 Leave a comment

Problem 1. Let {\Gamma} be the circumcircle of acute triangle {ABC}. Points {D} and {E} are on segments {AB} and {AC} respectively such that {AD = AE}. The perpendicular bisectors of {BD} and {CE} intersect minor arcs {AB} and {AC} of {\Gamma} at points {F} and {G} respectively. Prove that lines {DE} and {FG} are either parallel or they are the same line.

Problem 2. Find all integers {n \geq 3} for which there exist real numbers {a_1, a_2, \dots a_{n + 2}} satisfying {a_{n + 1} = a_1}, {a_{n + 2} = a_2} and

\displaystyle a_ia_{i + 1} + 1 = a_{i + 2}

For {i = 1, 2, \dots, n}.

Problem 3. An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from {1} to {10}

\displaystyle 4

\displaystyle 2\quad 6

\displaystyle 5\quad 7 \quad 1

\displaystyle 8\quad 3 \quad 10 \quad 9

Does there exist an anti-Pascal triangle with {2018} rows which contains every integer from {1} to {1 + 2 + 3 + \dots + 2018}?

Source: AoPS.

IMO 2016 – Problem 1

July 26, 2016 Leave a comment

IMO 2016, Problem 1. Triangle {BCF} has a right angle at {B}. Let {A} be the point on line {CF} such that {FA=FB} and {F} lies between {A} and {C}. Point {D} is chosen such that {DA=DC} and {AC} is the bisector of {\angle DAB}. Point {E} is chosen such that {EA=ED} and {AD} is the bisector of {\angle EAC}. Let {M} be the midpoint of {CF}. Let {X} be the point such that {AMXE} is a parallelogram (where {AM || EX} and {AE || MX}). Prove that the lines {BD, FX} and {ME} are concurrent.

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IMO 2015 Problem 1

July 10, 2015 Leave a comment

Problem 1. We say that a finite set {\mathcal{S}} of points in the plane is balanced if, for any two different points {A} and {B} in {\mathcal{S}}, there is a point {C} in {\mathcal{S}} such that {AC=BC}. We say that {\mathcal{S}} is center-free if for any three different points {A}, {B} and {C} in {\mathcal{S}}, there is no points {P} in {\mathcal{S}} such that {PA=PB=PC}.

(a) Show that for all integers {n\ge 3}, there exists a balanced set having {n} points.

(b) Determine all integers {n\ge 3} for which there exists a balanced center-free set having {n} points.

Problem 2. Find all triples of positive integers {(a, b, c)} such that {ab-c, bc-a, ca-b} are all powers of 2.

Problem 3. Let {ABC} be an acute triangle with {AB > AC}. Let {\Gamma } be its cirumcircle., {H} its orthocenter, and {F} the foot of the altitude from {A}. Let {M} be the midpoint of {BC}. Let {Q } be the point on { \Gamma } such that {\angle HQA } and let {K } be the point on {\Gamma } such that {\angle HKQ }. Assume that the points {A,B,C,K }and {Q } are all different and lie on {\Gamma} in this order.

Prove that the circumcircles of triangles {KQH } and {FKM } are tangent to each other.

Source: AoPS

Categories: Olympiad Tags: , ,

IMO 2014 Problem 6

July 9, 2014 1 comment

A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large {n}, in any set of {n} lines in general position it is possible to colour at least {\sqrt{n}} lines blue in such a way that none of its finite regions has a completely blue boundary.

Note: Results with {\sqrt{n}} replaced by {c\sqrt{n}} will be awarded points depending on the value of the constant {c}.

IMO 2014 Problem 6 (Day 2)

 

IMO 2014 Problem 5

July 9, 2014 1 comment

For every positive integer {n}, Cape Town Bank issues some coins that has {\frac{1}{n}} value. Let a collection of such finite coins (coins does not neccesarily have different values) which sum of their value is less than {99+\frac{1}{2}}. Prove that we can divide the collection into at most 100 groups such that sum of all coins’ value does not exceed 1.

IMO 2014 Problem 5 (Day 2)

Categories: Combinatorics, IMO, Olympiad Tags: ,

IMO 2014 Problem 4

July 9, 2014 Leave a comment

Let {P} and {Q} be on segment {BC} of an acute triangle {ABC} such that {\angle PAB=\angle BCA} and {\angle CAQ=\angle ABC}. Let {M} and {N} be the points on {AP} and {AQ}, respectively, such that {P} is the midpoint of {AM} and {Q} is the midpoint of {AN}. Prove that the intersection of {BM} and {CN} is on the circumference of triangle {ABC}.

IMO 2014 Problem 4 (Day 2)

Categories: Geometry, IMO, Olympiad Tags: ,
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