Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 23"

(solution)
(Standardized answer choices; minor edits)
 
(One intermediate revision by one other user not shown)
Line 2: Line 2:
 
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
 
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
  
−
A. <math>0</math>
+
<math>\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 2\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 1</math>
  
−
B. <math>5</math>
+
==Solution==
 +
Let <math>x</math> be the number of students in Math and Physics and <math>y</math> be the number of those in all three. Then, <math>4x</math> is the number of students in Math and Chem, and <math>3y</math> is the number of those in Physics and Chem.
  
−
C. <math>2</math>
+
Adding, <math>5x+4y=21</math>. We're looking for <math>y</math>, which is <math>4y\equiv21\equiv1\pmod{5}\Longrightarrow y\equiv4\pmod{5}</math>. Since <math>0\le4y<21</math>, the answer is <math>y=4\ \mathrm{(D)}</math>.
−
 
 
−
D. <math>4</math>
 
−
 
 
−
E. <math>1</math>
 
−
 
 
−
==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==

Latest revision as of 12:18, 26 April 2008

Problem

Of $21$ 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

$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 2\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 1$

Solution

Let $x$ be the number of students in Math and Physics and $y$ be the number of those in all three. Then, $4x$ is the number of students in Math and Chem, and $3y$ is the number of those in Physics and Chem.

Adding, $5x+4y=21$. We're looking for $y$, which is $4y\equiv21\equiv1\pmod{5}\Longrightarrow y\equiv4\pmod{5}$. Since $0\le4y<21$, the answer is $y=4\ \mathrm{(D)}$.

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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_Cyprus_MO/Lyceum/Problem_23&oldid=25562"