Difference between revisions of "2024 AMC 10B Problems/Problem 12"
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| + | ==Problem== | ||
| + | A group of <math>100</math> students from different countries meet at a mathematics competition. | ||
| + | Each student speaks the same number of languages, and, for every pair of | ||
| + | students <math>A</math> and <math>B</math>, student <math>A</math> speaks some language that student <math>B</math> does not speak, | ||
| + | and student <math>B</math> speaks some language that student <math>A</math> does not speak. What is the | ||
| + | least possible total number of languages spoken by all the students? | ||
| + | |||
| + | <math>\textbf{(A) } 9 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 51 \qquad\textbf{(E) } 100</math> | ||
| + | |||
==Solution 1== | ==Solution 1== | ||
Let's say we have some number of languages. Then each student will speak some amount of those languages, and no two people can have the same combination of languages or else the conditions will no longer be satisfied. Notice that <math>{9}\choose{4}</math> <math>= 126 \geq 100</math>. So each of the <math>100</math> students can speak some <math>4</math> of the <math>9</math> languages. Thus, <math>\boxed{9}</math> is our answer. | Let's say we have some number of languages. Then each student will speak some amount of those languages, and no two people can have the same combination of languages or else the conditions will no longer be satisfied. Notice that <math>{9}\choose{4}</math> <math>= 126 \geq 100</math>. So each of the <math>100</math> students can speak some <math>4</math> of the <math>9</math> languages. Thus, <math>\boxed{9}</math> is our answer. | ||
~lprado | ~lprado | ||
| + | |||
| + | ==See also== | ||
| + | {{AMC10 box|year=2024|ab=B|num-b=11|num-a=13}} | ||
| + | {{MAA Notice}} | ||
Revision as of 00:59, 14 November 2024
Problem
A group of 
students from different countries meet at a mathematics competition.
Each student speaks the same number of languages, and, for every pair of
students 
and 
, student speaks some language that student does not speak,
and student speaks some language that student does not speak. What is the
least possible total number of languages spoken by all the students?
Solution 1
Let's say we have some number of languages. Then each student will speak some amount of those languages, and no two people can have the same combination of languages or else the conditions will no longer be satisfied. Notice that 
. So each of the students can speak some 
of the 
languages. Thus, 
is our answer.
~lprado
See also
| 2024 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 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 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
