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Return a single integer only. The problem has been revised so that no final answer is an irrational number, a parameterized expression, a range, a proof, a drawing, a graph, a locus equation, or a region description.
Let , and define
For every integer , define the periodic triangular-wave function by
Find the number of intersection points of and in the range .
With a=3/5:
f(x) = (3/40)(x-1)² + 10/3 - 3 = (3/40)(x-1)² + 1/3
The minimum of f is at x=1, value 1/3.
g(x) is a triangular wave oscillating between 0 and 1 with period 2.
We need intersections of f with g for x≥0.
f(x) ≥ 1/3, increasing parabola away from x=1. g ranges in [0,1].
Find where f(x) ≤ 1: (3/40)(x-1)² + 1/3 ≤ 1 (3/40)(x-1)² ≤ 2/3 (x-1)² ≤ (2/3)(40/3) = 80/9 ≈ 8.889 |x-1| ≤ 2.981 So x ∈ [1-2.981, 1+2.981] = [-1.981, 3.981]
For x≥0: x ∈ [0, 3.981]. Beyond x≈3.981, f>1>g, no intersections.
So intersections only in [0, 3.981].
Let me evaluate f at key points:
- f(0) = (3/40)(1) + 1/3 = 0.075 + 0.333 = 0.408
- f(1) = 1/3 = 0.333
- f(2) = 0.075 + 0.333 = 0.408
- f(3) = (3/40)(4) + 1/3 = 0.3 + 0.333 = 0.633
- f(4) = (3/40)(9)+1/3 = 0.675+0.333 = 1.008
g values:
- g(0)=0, g(1)=1, g(2)=0, g(3)=1, g(4)=0
Let me analyze each segment.
Segment [0,1]: g(x)=x (rising from 0 to 1). f decreases from 0.408 to 0.333. At x=0: g=0, f=0.408, f>g. At x=1: g=1, f=0.333, f<g. One crossing. 1 intersection.
Segment [1,2]: g(x)=-x+2 (falling from 1 to 0). f rises from 0.333 to 0.408. At x=1: g=1, f=0.333, f<g. At x=2: g=0, f=0.408, f>g. One crossing. 1 intersection.
Segment [2,3]: g(x)=x-2 (rising 0 to 1). f rises from 0.408 to 0.633. At x=2: g=0, f=0.408, f>g. At x=3: g=1, f=0.633, f<g. One crossing. 1 intersection.
Segment [3,4]: g(x)=-x+4 (falling 1 to 0). f rises from 0.633 to 1.008. At x=3: g=1, f=0.633, f<g. At x=4: g=0, f=1.008, f>g. One crossing. 1 intersection.
Segment [4,5]: g(x)=x-4 (rising 0 to 1). f rises from 1.008 upward. At x=4: f=1.008 > g=0. f(4.5) = (3/40)(12.25)+1/3 = 0.919+0.333=1.252, g=0.5, f>g. f(5)=(3/40)(16)+1/3=1.2+0.333=1.533>1. f>g always. 0.
Beyond x=4, f>1, no more intersections.
Total: 1+1+1+1 = 4
\boxed{4}
Comparative Analysis (P28) (Average across runs)
Averages use reported measurements only. Missing usage is not treated as zero.