Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1993 based on Integers.
Find the number of four topics of integers (a,b,c,d) with 0<a<b<c<d<500 satisfy a+d=b+c and bc-ad=93.
Integers
Digits
Algebra
Answer: is 870.
AIME I, 1993, Question 4
Elementary Algebra by Hall and Knight
Let k=a+d=b+c
or, d=k-a, b=k-c,
or, (k-c)c-a(k-a)=k(c-a)-(c-a)(c+a)
=(a-c)(a+c-k)
=(c-a)(d-c)=93
(c-a)(d-c)=(1,93),(3,31),(31,3),(93,1)
solving for c
(a,b,c,d)=(c-93,c-92,c,c+1),(c-31,c-28,c,c+3),(c-1,c+92,c,c+93),(c-3,c+28,c,c+31)
taking first two solutions a<b<c<d<500
or,\(1 \leq c-93, c+1 \leq 499\)
or, \(94 \leq c \leq 498 \) gives 405 solutions
and \(1 \leq c-31, c+3 \leq 499\)
or, \(32 \leq c \leq 496\) gives 465 solutions
or, 405+465=870 solutions.
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.