MathDB

2003 IMO Shortlist

Part of IMO Shortlist

Subcontests

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Sequence inequality

Let nn be a positive integer and let (x1,…,xn)(x_1,\ldots,x_n), (y1,…,yn)(y_1,\ldots,y_n) be two sequences of positive real numbers. Suppose (z2,…,z2n)(z_2,\ldots,z_{2n}) is a sequence of positive real numbers such that zi+j2≥xiyjz_{i+j}^2 \geq x_iy_j for all 1≤i,j≤n1\le i,j \leq n.
Let M=max⁡{z2,…,z2n}M=\max\{z_2,\ldots,z_{2n}\}. Prove that (M+z2+⋯+z2n2n)2≥(x1+⋯+xnn)(y1+⋯+ynn). \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right).
[hide="comment"] Edited by Orl.
Proposed by Reid Barton, USA

Nice and hard "nt" problem on digital representati

Let f(k)f(k) be the number of integers nn satisfying the following conditions:
(i) 0≤n<10k0\leq n < 10^k so nn has exactly kk digits (in decimal notation), with leading zeroes allowed;
(ii) the digits of nn can be permuted in such a way that they yield an integer divisible by 1111.
Prove that f(2m)=10f(2m−1)f(2m) = 10f(2m-1) for every positive integer mm.
Proposed by Dirk Laurie, South Africa
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A beautiful sequence with a nice property!

The sequence a0a_0, a1a_1, a2,a_2, …\ldots is defined as follows: a_0=2, \qquad a_{k+1}=2a_k^2-1  \text{for }k \geq 0. Prove that if an odd prime pp divides ana_n, then 2n+32^{n+3} divides p2−1p^2-1.
[hide="comment"] Hi guys ,
Here is a nice problem:
Let be given a sequence ana_n such that a0=2a_0=2 and an+1=2an2−1a_{n+1}=2a_n^2-1 . Show that if pp is an odd prime such that p∣anp|a_n then we have p2≡1(mod2n+3)p^2\equiv 1\pmod{2^{n+3}}
Here are some futher question proposed by me :Prove or disprove that : 1) gcd(n,an)=1gcd(n,a_n)=1 2) for every odd prime number pp we have am≡±1(modp)a_m\equiv \pm 1\pmod{p} where m=p2−12km=\frac{p^2-1}{2^k} where k=1k=1 or 22
Thanks kiu si u
Edited by Orl.

IMO ShortList 2003, geometry problem 7

Let ABCABC be a triangle with semiperimeter ss and inradius rr. The semicircles with diameters BCBC, CACA, ABAB are drawn on the outside of the triangle ABCABC. The circle tangent to all of these three semicircles has radius tt. Prove that s2<t≤s2+(1−32)r.\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. Alternative formulation. In a triangle ABCABC, construct circles with diameters BCBC, CACA, and ABAB, respectively. Construct a circle ww externally tangent to these three circles. Let the radius of this circle ww be tt. Prove: s2<t≤s2+12(2−3)r\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r, where rr is the inradius and ss is the semiperimeter of triangle ABCABC.
Proposed by Dirk Laurie, South Africa
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IMO ShortList 2003, number theory problem 1

Let mm be a fixed integer greater than 11. The sequence x0x_0, x1x_1, x2x_2, …\ldots is defined as follows: xi={2iif 0≤i≤m−1;∑j=1mxi−jif i≥m.x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases} Find the greatest kk for which the sequence contains kk consecutive terms divisible by mm .
Proposed by Marcin Kuczma, Poland

Imo shortlist 2003, algebra problem 1

Let aija_{ij} i=1,2,3i=1,2,3; j=1,2,3j=1,2,3 be real numbers such that aija_{ij} is positive for i=ji=j and negative for i≠ji\neq j.
Prove the existence of positive real numbers c1c_{1}, c2c_{2}, c3c_{3} such that the numbers a11c1+a12c2+a13c3,a21c1+a22c2+a23c3,a31c1+a32c2+a33c3a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3} are either all negative, all positive, or all zero.
Proposed by Kiran Kedlaya, USA
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IMO ShortList 2003, geometry problem 4

Let Γ1\Gamma_1, Γ2\Gamma_2, Γ3\Gamma_3, Γ4\Gamma_4 be distinct circles such that Γ1\Gamma_1, Γ3\Gamma_3 are externally tangent at PP, and Γ2\Gamma_2, Γ4\Gamma_4 are externally tangent at the same point PP. Suppose that Γ1\Gamma_1 and Γ2\Gamma_2; Γ2\Gamma_2 and Γ3\Gamma_3; Γ3\Gamma_3 and Γ4\Gamma_4; Γ4\Gamma_4 and Γ1\Gamma_1 meet at AA, BB, CC, DD, respectively, and that all these points are different from PP. Prove that AB⋅BCAD⋅DC=PB2PD2. \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.

When 11...1 22...2 5 is a perfect square

Let b b be an integer greater than 5 5. For each positive integer n n, consider the number x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, written in base b b.
Prove that the following condition holds if and only if b \equal{} 10: there exists a positive integer M M such that for any integer n n greater than M M, the number xn x_n is a perfect square.
Proposed by Laurentiu Panaitopol, Romania

IMO ShortList 2003, combinatorics problem 4

Let x1,…,xnx_1,\ldots, x_n and y1,…,yny_1,\ldots, y_n be real numbers. Let A=(aij)1≤i,j≤nA = (a_{ij})_{1\leq i,j\leq n} be the matrix with entries aij={1,if xi+yj≥0;0,if xi+yj<0.a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases} Suppose that BB is an n×nn\times n matrix with entries 00, 11 such that the sum of the elements in each row and each column of BB is equal to the corresponding sum for the matrix AA. Prove that A=BA=B.
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triangle geometry [AP^2 + PD^2 = ...]

Let ABCABC be a triangle and let PP be a point in its interior. Denote by DD, EE, FF the feet of the perpendiculars from PP to the lines BCBC, CACA, ABAB, respectively. Suppose that AP2+PD2=BP2+PE2=CP2+PF2.AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2. Denote by IAI_A, IBI_B, ICI_C the excenters of the triangle ABCABC. Prove that PP is the circumcenter of the triangle IAIBICI_AI_BI_C.
Proposed by C.R. Pranesachar, India

Calculus rather than inequalities

Consider pairs of the sequences of positive real numbers a1≥a2≥a3≥⋯ ,b1≥b2≥b3≥⋯a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots and the sums A_n = a_1 + \cdots + a_n,  B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots. For any pair define cn=min⁡{ai,bi}c_n = \min\{a_i,b_i\} and Cn=c1+⋯+cnC_n = c_1 + \cdots + c_n, n=1,2,…n=1,2,\ldots.
(1) Does there exist a pair (ai)i≥1(a_i)_{i\geq 1}, (bi)i≥1(b_i)_{i\geq 1} such that the sequences (An)n≥1(A_n)_{n\geq 1} and (Bn)n≥1(B_n)_{n\geq 1} are unbounded while the sequence (Cn)n≥1(C_n)_{n\geq 1} is bounded?
(2) Does the answer to question (1) change by assuming additionally that bi=1/ib_i = 1/i, i=1,2,…i=1,2,\ldots?
Justify your answer.

A bit of geometry, combinatorics, and number theory

Let n≥5n \geq 5 be a given integer. Determine the greatest integer kk for which there exists a polygon with nn vertices (convex or not, with non-selfintersecting boundary) having kk internal right angles.
Proposed by Juozas Juvencijus Macys, Lithuania