PROBLEMS AND SOLUTIONS Problems and Solutions Baltic Way 2012 which holds, since x2 +y2 +xy ≥0 for all real numbers x, y. Remark. The inequality k2 + l2 + kl −3l −3k + 3 ≥0 can also be proven by separating perfect squares as 1 4 (k −l)2 + 3 4 (k +l)2 −3·(k +l)+ 3 ≥0 which is in turn

similar to

BALTIC WAY 1993

Baltic Way 1993 Riga, November 13, 1993 Problems and solutions 1. a1a2a3 and a3a2a1 are two three-digit decimal numbers, with a1, a3 being different non-zero digits. The squares of these numbers are five-digit numbers b1b2b3b4b5 and b5b4b3b2b1 respectively. Find all such three-digit numbers. THIRD INTERNATIONAL OLYMPIAD, 1961 Third International Olympiad, 1961 1961/1. Solve the system of equations: x+y +z = a x2 +y 2+z = b2 xy = z2 where a and b are constants. Give the conditions that a and b must satisfy so that x;y;z (the solutions of the system) are distinct positive numbers. 1961/2. Let a;b;c be the sides of a triangle, and T its area. Prove: a2+b2+c2 ‚ 4 p 3T: In what case does equality hold?

CONTENTS

Solutions Baltic Way 2014 Problem 5 Given positive real numbers a, b, c, dthat satisfy equalities a 2+ d 22ad= b 2+ c + bc and a + b = c + d2, nd all possible values

BALTIC WAY 2017

Baltic Way 2017 Sorø, November 11th, 2017 Time allowed: 4.5 hours. During the first 30 minutes, questions may be asked. Tools for writing and drawing are the only ones allowed.

BALTIC WAY 1995

Baltic Way 1995 V¨aster˚as (Sweden), November 12, 1995 Problems and solutions 1. Find all triples (x,y,z) of positive integers satisfying the system of equations

E 11 D C 17 A B

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ABELKONKURRANSEN

* +-,/.1032 465 784 9;:8:8: =@?BA6CEDF=HGJILK MON)PRQSILK>TVU6U6WYXZQSI\ ?^ILD =H_`_^Da=HbOcedfUgTSIL_`W h i ikjBlmBnpo lrqbadc e CgfPfW> + , ,.-0/102 h i c Cgf j;klT Y m n fPol>A@ Yqp k?T Y PROBLEMS AND SOLUTIONS Problems and Solutions Baltic Way 2012 which holds, since x2 +y2 +xy ≥0 for all real numbers x, y. Remark. The inequality k2 + l2 + kl −3l −3k + 3 ≥0 can also be proven by separating perfect squares as 1 4 (k −l)2 + 3 4 (k +l)2 −3·(k +l)+ 3 ≥0 which is in turn

similar to

BALTIC WAY 1993

Baltic Way 1993 Riga, November 13, 1993 Problems and solutions 1. a1a2a3 and a3a2a1 are two three-digit decimal numbers, with a1, a3 being different non-zero digits. The squares of these numbers are five-digit numbers b1b2b3b4b5 and b5b4b3b2b1 respectively. Find all such three-digit numbers. THIRD INTERNATIONAL OLYMPIAD, 1961 Third International Olympiad, 1961 1961/1. Solve the system of equations: x+y +z = a x2 +y 2+z = b2 xy = z2 where a and b are constants. Give the conditions that a and b must satisfy so that x;y;z (the solutions of the system) are distinct positive numbers. 1961/2. Let a;b;c be the sides of a triangle, and T its area. Prove: a2+b2+c2 ‚ 4 p 3T: In what case does equality hold?

CONTENTS

Solutions Baltic Way 2014 Problem 5 Given positive real numbers a, b, c, dthat satisfy equalities a 2+ d 22ad= b 2+ c + bc and a + b = c + d2, nd all possible values

BALTIC WAY 2017

Baltic Way 2017 Sorø, November 11th, 2017 Time allowed: 4.5 hours. During the first 30 minutes, questions may be asked. Tools for writing and drawing are the only ones allowed.

BALTIC WAY 1995

Baltic Way 1995 V¨aster˚as (Sweden), November 12, 1995 Problems and solutions 1. Find all triples (x,y,z) of positive integers satisfying the system of equations

E 11 D C 17 A B

X~·L67ID:A`N_9S\S@_9bf8;rD_TSYit25SM:H_>¸254 :[a?FA4d25FA>M4?b it25F#^tgW25F.uce8:mnw& b wnxvf²;zw&7 -