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Balkan Mathematical Olympiad 2018

June 23, 2018 Leave a comment

Problem 1. A quadrilateral {ABCD} is inscribed in a circle {k}, where {AB>CD} and {AB} is not parallel to {CD}. Point {M} is the intersection of the diagonals {AC} and {BD} and the perpendicular from {M} to {AB} intersects the segment {AB} at the point {E}. If {EM} bisects the angle {CED}, prove that {AB} is a diameter of the circle {k}.

Problem 2. Let {q} be a positive rational number. Two ants are initially at the same point {X} in the plane. In the {n}-th minute {(n=1,2,...)} each of them chooses whether to walk due north, east, south or west and then walks the distance of {q^n} meters. After a whole number of minutes, they are at the same point in the plane (non necessarily {X}), but have not taken exactly the same route within that time. Determine all the possible values of {q}.

Problem 3. Alice and Bob play the following game: They start with two non-empty piles of coins. Taking turns, with Alice playing first, each player chooses a pile with an even number of coins and moves half of the coins of this pile to the other pile. The came ends if a player cannot move, in which case the other player wins.

Determine all pairs {(a,b)} of positive integers such that if initially the two piles have {a} and {b} coins, respectively, then Bob has a winning strategy.

Problem 4. Find all primes {p} and {q} such that {3p^{q-1}+1} divides {11^q+17^p}.

Source: https://bmo2018.dms.rs/wp-content/uploads/2018/05/BMOproblems2018_English.pdf

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Balkan Mathematical Olympiad 2017 – Problems

May 10, 2017 Leave a comment

Problem 1. Find all ordered pairs of positive integers { (x, y)} such that:

\displaystyle x^3+y^3=x^2+42xy+y^2.

Problem 2. Consider an acute-angled triangle {ABC} with {AB<AC} and let {\omega} be its circumscribed circle. Let {t_B} and {t_C} be the tangents to the circle {\omega} at points {B} and {C}, respectively, and let {L} be their intersection. The straight line passing through the point {B} and parallel to {AC} intersects {t_C} in point {D}. The straight line passing through the point {C} and parallel to {AB} intersects {t_B} in point {E}. The circumcircle of the triangle {BDC} intersects {AC} in {T}, where {T} is located between {A} and {C}. The circumcircle of the triangle {BEC} intersects the line {AB} (or its extension) in {S}, where {B} is located between {S} and {A}.

Prove that {ST}, {AL}, and {BC} are concurrent.

Problem 3. Let {\mathbb{N}} denote the set of positive integers. Find all functions {f:\mathbb{N}\longrightarrow\mathbb{N}} such that

\displaystyle n+f(m)\mid f(n)+nf(m)

for all {m,n\in \mathbb{N}}

Problem 4. On a circular table sit {\displaystyle {n> 2}} students. First, each student has just one candy. At each step, each student chooses one of the following actions:

  • (A) Gives a candy to the student sitting on his left or to the student sitting on his right.
  • (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right.

At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions.

(Two distributions are different if there is a student who has a different number of candy in each of these distributions.)

Source: AoPS

Balkan Mathematical Olympiad – 2016 Problems

Problem 1. Find all injective functions {f: \mathbb R \rightarrow \mathbb R} such that for every real number {x} and every positive integer {n},

\displaystyle \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016

Problem 2. Let {ABCD} be a cyclic quadrilateral with {AB<CD}. The diagonals intersect at the point {F} and lines {AD} and {BC} intersect at the point {E}. Let {K} and {L} be the orthogonal projections of {F} onto lines {AD} and {BC} respectively, and let {M}, {S} and {T} be the midpoints of {EF}, {CF} and {DF} respectively. Prove that the second intersection point of the circumcircles of triangles {MKT} and {MLS} lies on the segment {CD}.

Problem 3. Find all monic polynomials {f} with integer coefficients satisfying the following condition: there exists a positive integer {N} such that {p} divides {2(f(p)!)+1} for every prime {p>N} for which {f(p)} is a positive integer.

Problem 4. The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of {1201} colours so that no rectangle with perimeter {100} contains two squares of the same colour. Show that no rectangle of size {1\times1201} or {1201\times1} contains two squares of the same colour.

Balkan Mathematical Olympiad 2013

June 30, 2013 Leave a comment

Problem 1. In a triangle {ABC}, the excircle {\omega_a} opposite to {A} touches {AB} at {P} and {AC} at {Q} and the excircle {\omega_b} opposite to {B} touches {BA} at {M} and {BC} at {N}. Let {K} be the projection of {C} onto {MN}, and let {L} be the projection of {C} onto {PQ}.

Show that the quadrilateral {MKLP} is cyclic.

Problem 2. Determine all positive integers {x,y,z} such that

\displaystyle x^5+4^y=2013^z.

Problem 3. Let {S} be the set of positive real numbers. Find all functions {f:S^3 \rightarrow S} such that for all positive real numbers {x,y,z} and {k} the following three conditions are satisfied:

  • (a) {xf(x,y,z)=zf(z,y,x)},
  • (b) {f(x,yk,k^2z)=kf(x,y,z)},
  • (c) {f(1,k,k+1)=k+1}.

Problem 4. In a mathematical competition some competitors are friends; friendship is always mutual, that is to say that when {A} is a friend of {B}, then also {B} is a friend of {A}. We say that {n \geq 3} different competitors {A_1,A_2,..,A_n} form a weakly-friendly cycle if {A_i} is not a friend of {A_{i+1}} for {1 \leq i \leq n (A_{n+1}=A_1)}, and there are no other pairs of non-friends among the components of this cycle.

The following property is satisfied: for every competitor {C}, and every weakly-friendly cycle {\mathcal{S}} of competitors not including {C}, the set of competitors {D} in {\mathcal{S}} which are not friends of {C} has at most one element.

Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors that are in the same room are friends.

Balkan Mathematical Olympiad 2011 Problem 4

May 14, 2011 Leave a comment

Let ABCDEF be a convex hexagon of area 1, whose opposite sides are parallel. The lines AB, CD and EF meet in pairs to determine the vertices of a triangle. Similarly, the lines BC, DE and FA meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least 3/2.

BMO 2011 Problem 4

Balkan Mathematical Olympiad 2011 Problem 3

May 14, 2011 Leave a comment

Let S be a finite set of positive integers which has the following property: if x is a member of S,then so are all positive divisors of x. A non-empty subset T of S is good if whenever x,y\in T and x<y, the ratio y/x is a power of a prime number. A non-empty subset T of S is bad if whenever x,y\in T and x<y, the ratio y/x is not a power of a prime number. A set of an element is considered both good and bad. Let k be the largest possible size of a good subset of S. Prove that k is also the smallest number of pairwise-disjoint bad subsets whose union is S.

BMO 2011 Problem 3

Categories: Combinatorics, IMO, Olympiad Tags: , ,

Balkan Mathematical Olympiad 2011 Problem 2

May 14, 2011 2 comments

Given real numbers x,y,z such that x+y+z=0, show that
\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0
When does equality hold?

BMO 2011 Problem 2

Categories: IMO, Inequalities, Olympiad Tags: , ,
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