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2014 Bosnia Herzegovina Team Selection Test

The document is a test for selection to the 2014 Bosnia Herzegovina national soccer team. It contains 6 math and geometry problems to be solved over 2 days. Day 1 has 3 problems involving properties of circles, expressions with rational functions of real numbers, and finding integer solutions to an equation. Day 2 has 3 problems about determining a value for the sum of terms in a sequence, counting the number of triangles that can be formed inside a regular polygon using its diagonals, and proving a relationship between points in a triangle given a condition about line segments.

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JHEFFERSON JESUS LOPEZ QUISPE
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166 views1 page

2014 Bosnia Herzegovina Team Selection Test

The document is a test for selection to the 2014 Bosnia Herzegovina national soccer team. It contains 6 math and geometry problems to be solved over 2 days. Day 1 has 3 problems involving properties of circles, expressions with rational functions of real numbers, and finding integer solutions to an equation. Day 2 has 3 problems about determining a value for the sum of terms in a sequence, counting the number of triangles that can be formed inside a regular polygon using its diagonals, and proving a relationship between points in a triangle given a condition about line segments.

Uploaded by

JHEFFERSON JESUS LOPEZ QUISPE
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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2014 Bosnia Herzegovina Team Selection Test

Bosnia Herzegovina Team Selection Test 2014

Day 1 May 10th

1 Let k be the circle and A and B points on circle which are not diametrically
opposite. On minor arc AB lies point arbitrary point C. Let D, E and F be
foots of perpendiculars from √
C on chord AB and tangents of circle k in points
A and B. Prove that CD = CE · CF

1+ab 1+bc
2 Let a ,b and c be distinct real numbers. a) Determine value of a−b · b−c +
1+bc 1+ca 1+ca 1+ab
b−c · c−a + c−a · a−b
1−ab 1−bc 1−bc 1−ca 1−ca 1−ab
b) Determine value of a−b · b−c + b−c · c−a + c−a · a−b
1+a2 b2 1+b2 c2 1+c2 a2 3
c) Prove the following ineqaulity (a−b)2
+ (b−c)2
+ (c−a)2
≥ 2
When does eqaulity holds?

3 Find all nonnegative integer numbers such that 7x − 2 · 5y = −1

Day 2 May 11th

am−1
1 Sequence an is defined by a1 = 21 , am = 2m·a m−1 +1
for m > 1. Determine value
of a1 + a2 + ... + ak in terms of k, where k is positive integer.

2 It is given regular n-sided polygon, n ≥ 6. How many triangles they are inside
the polygon such that all of their sides are formed by diagonals of polygon and
their vertices are vertices of polygon?

3 Let D and E be foots of altitudes from A and B of triangle ABC, F be


intersection point of angle bisector from C with side AB, and O, I and H
be circumcenter, center of inscribed circle and orthocenter of triangle ABC,
CF CF
respectively. If AD + BE = 2, prove that OI = IH.

www.artofproblemsolving.com/community/c3666
Contributors: gobathegreat

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