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Romanian Masters in Mathematics contest – 2018

March 9, 2018 Leave a comment

Problem 1. Let {ABCD} be a cyclic quadrilateral an let {P} be a point on the side {AB.} The diagonals {AC} meets the segments {DP} at {Q.} The line through {P} parallel to {CD} mmets the extension of the side {CB} beyond {B} at {K.} The line through {Q} parallel to {BD} meets the extension of the side {CB} beyond {B} at {L.} Prove that the circumcircles of the triangles {BKP} and {CLQ} are tangent .

Problem 2. Determine whether there exist non-constant polynomials {P(x)} and {Q(x)} with real coefficients satisfying

\displaystyle P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.

Problem 3. Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

Problem 4. Let {a,b,c,d} be positive integers such that {ad \neq bc} and {gcd(a,b,c,d)=1}. Let {S} be the set of values attained by {\gcd(an+b,cn+d)} as {n} runs through the positive integers. Show that {S} is the set of all positive divisors of some positive integer.

Problem 5. Let {n} be positive integer and fix {2n} distinct points on a circle. Determine the number of ways to connect the points with {n} arrows (oriented line segments) such that all of the following conditions hold:

  • each of the {2n} points is a startpoint or endpoint of an arrow;
  • no two arrows intersect;
  • there are no two arrows {\overrightarrow{AB}} and {\overrightarrow{CD}} such that {A}, {B}, {C} and {D} appear in clockwise order around the circle (not necessarily consecutively).

Problem 6. Fix a circle {\Gamma}, a line {\ell} to tangent {\Gamma}, and another circle {\Omega} disjoint from {\ell} such that {\Gamma} and {\Omega} lie on opposite sides of {\ell}. The tangents to {\Gamma} from a variable point {X} on {\Omega} meet {\ell} at {Y} and {Z}. Prove that, as {X} varies over {\Omega}, the circumcircle of {XYZ} is tangent to two fixed circles.

Source: Art of Problem Solving forums

Some quick ideas: For Problem 1 just consider the intersection of the circle {(BKP)} with the circle {(ABCD)}. You’ll notice immediately that this point belongs to the circle {(CLQ)}. Furthermore, there is a common tangent to the two circles at this point.

For Problem 2 we have {10\deg P = 21 \deg Q}. Eliminate the highest order term from both sides and look at the next one to get a contradiction.

Problem 4 becomes easy after noticing that if {q} divides {an+b} and {cn+d} then {q} divides {ad-bc}.

In Problem 5 try to prove that the choice of start points determines that of the endpoints. Then you have a simple combinatorial proof.

Problem 6 is interesting and official solutions use inversions. Those are quite nice, but it may be worthwhile to understand what happens in the non-inverted configuration.

I will come back to some of these problems in some future posts.

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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

Missing digit – short puzzle

April 15, 2017 Leave a comment

The number 2^{29} has 9 digits, all different; which digit is missing?

Mathematical Mind-Benders, Peter Winkler

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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: , ,

Monotonic bijection from naturals to pairs of natural numbers

October 23, 2014 Leave a comment

This is a cute problem I found this evening.

Suppose {\phi : \Bbb{N}^* \rightarrow \Bbb{N}^*\times \Bbb{N}^*} is a bijection such that if {\phi(k) = (i,j),\ \phi(k')=(i',j')} and {k \leq k'}, then {ij \leq i'j'}.

Prove that if {k = \phi(i,j)} then {k \geq ij}.

Proof: The trick is to divide the pairs of positive integers into families with the same product.

\displaystyle \begin{matrix} (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) & \cdots \\ & (2,1) & (3,1) & (2,2) & (5,1) & (2,3) & \cdots \\ & & &( 4,1) & & (3,2) & \cdots \\ & & & & & (6,1) & \cdots \end{matrix}

Note that the {M}-th column contains as many elements as the number of divisors of {M}. Now we just just use a simple observation. Let {\phi(k)=(i,j)} be on the {M}-th column (i.e. {ij = M}). If {n \geq 1} then {\phi(k+n)=(i',j')} cannot be on one of the first {M-1} columns. Indeed, the monotonicity property implies {M = ij \leq i'j'}. The fact that {\phi} is a bijection assures us that {\phi(1),...,\phi(k)} cover the first {M-1} columns. Moreover, one element from the {M}-th column is surely covered, namely {(i,j) = \phi(k)}. This means that

\displaystyle k \geq d(1)+...+d(M-1)+1 \geq M = ij,

where we have denoted by {d(n)} the number of positive divisors of {n}.

IMC 2014 Day 1 Problem 4

July 31, 2014 2 comments

Let {n > 6} be a perfect number and {n = p_1^{e_1}...p_k^{e_k}} be its prime factorization with {1<p_1<...<p_k}. Prove that {e_1} is an even number.

A number {n} is perfect if {s(n)=2n}, where {s(n)} is the sum of the divisors of {n}.

IMC 2014 Day 1 Problem 4

Group actions and applications to number theory

April 6, 2014 Leave a comment

Theorem. Let {G} be a finite group and {p} be a prime number. Then {|G|^{p-1} \equiv |\{x \in G : x^p=e\}|}. (we denote by {|X|} the cardinal of {X}).

Proof: Denote {X = \{(g_1,...,g_p) \in \Bbb{Z}^p : g_1...g_p=e\}}. Then {|X|=|G|^{n-1}} since if we cnoose the first {p-1} elements arbitrarily in {G} then the last element can be chosen in a unique way such that the equation is satisfied. Note that {X} is stable under cyclic permutations of the elements of a vector. Therefore the action of {\Bbb{Z}_p} on {X} by

\displaystyle j\cdot (g_1,...,g_p)=(g_{j+1},...,g_{j+p})

is well defined.

We will need the following lemma:

Lemma. If {G} is a {p} group (with {p} prime) and {G} acts on {X} then

\displaystyle |X| \equiv |\{x \in X : G\cdot x = \{x\}\}| \pmod p.

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