Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 23"
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==Problem== | ==Problem== | ||
| − | + | Of <math>21</math> students taking Mathematics, Physics and Chemistry, no student takes one subject only. The number of students taking Mathematics and Chemistry only, equals to four times the number taking Mathematics and Physics only. If the number of students taking Physics and Chemistry only equals to three times the number of students taking all three subjects, then the number of students taking all three subjects is | |
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| + | A. <math>0</math> | ||
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| + | B. <math>5</math> | ||
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| + | C. <math>2</math> | ||
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| + | D. <math>4</math> | ||
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| + | E. <math>1</math> | ||
==Solution== | ==Solution== | ||
| − | {{ | + | Let <math>x =</math> number of students in Math and Physics, <math>4x = </math> number of students in Math and Chem, <math>y = </math> those in all three, <math>3y = </math>those in Physics and Chem. Adding, <math>5x + 4y = 21</math>. We're looking for <math>y</math>, which is <math>4y \equiv 21 \equiv 1 \pmod{5} \Longrightarrow y \equiv 4 \pmod{5}</math>. Since <math>0 \le 4y < 21</math>, the answer is <math>y = 4\ \mathrm{D}</math>. |
==See also== | ==See also== | ||
{{CYMO box|year=2006|l=Lyceum|num-b=22|num-a=24}} | {{CYMO box|year=2006|l=Lyceum|num-b=22|num-a=24}} | ||
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| + | [[Category:Introductory Algebra Problems]] | ||
Revision as of 17:14, 15 October 2007
Problem
Of 
students taking Mathematics, Physics and Chemistry, no student takes one subject only. The number of students taking Mathematics and Chemistry only, equals to four times the number taking Mathematics and Physics only. If the number of students taking Physics and Chemistry only equals to three times the number of students taking all three subjects, then the number of students taking all three subjects is
A. 
B. 
C. 
D. 
E. 
Solution
Let 
number of students in Math and Physics, 
number of students in Math and Chem, 
those in all three, 
those in Physics and Chem. Adding, 
. We're looking for 
, which is 
. Since 
, the answer is 
.
See also
| 2006 Cyprus MO, Lyceum (Problems) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
