AMC 10A 2026 (Problem 14)Consider a triangle with positive area whose side lengths are functions of ttt given by a=t2+1a=t^2+1a=t2+1, b=5t+7b=5t+7b=5t+7, and c=4t+4c=4t+4c=4t+4. How many different integer values of ttt are possible?(A) 0\text{(A)}\;0(A)0(B) 7\text{(B)}\;7(B)7(C) 9\text{(C)}\;9(C)9(D) 10\text{(D)}\;10(D)10(E) infinitely many\text{(E)}\;\text{infinitely many}(E)infinitely manyRelated TopicsCoreTriangle InequalityQuadratic InequalitiesCheck AnswerYour answer:ABCDECheckHints (5)Hint 1By the Triangle Inequality,a+b>ca+b>ca+b>ca+c>ba+c>ba+c>bb+c>ab+c>ab+c>aHint 2a+b>ca+b>ca+b>ct2+1+5t+7>4t+4t^2+1+5t+7>4t+4t2+1+5t+7>4t+4t2+t+4>0t^2+t+4>0t2+t+4>0(t+12)2+154>0✓\left(t+\frac12\right)^2+\frac{15}{4}>0\quad\checkmark(t+21)2+415>0✓Hint 3a+c>ba+c>ba+c>bt2+1+4t+4>5t+7t^2+1+4t+4>5t+7t2+1+4t+4>5t+7t2−t−2>0t^2-t-2>0t2−t−2>0(t−2)(t+1)>0(t-2)(t+1)>0(t−2)(t+1)>0t>2ort<−1t>2\quad\text{or}\quad t<-1t>2ort<−1Hint 4b+c>ab+c>ab+c>a5t+7+4t+4>t2+15t+7+4t+4>t^2+15t+7+4t+4>t2+10>t2−9t−100>t^2-9t-100>t2−9t−100>(t−10)(t+1)0>(t-10)(t+1)0>(t−10)(t+1)−1<t<10-1<t<10−1<t<10t=0,1,2,…,9t=0,1,2,\ldots,9t=0,1,2,…,9Hint 5By Hints 3 and 4,t=3,4,5,6,7,8,9t=3,4,5,6,7,8,9t=3,4,5,6,7,8,9Ans=7\text{Ans}=7Ans=7Final Answer(B) 7Related Problems (5)AMC 10A 2025 (Problem 3)AMC 12B 2024 (Problem 24)AMC 10B 2023 (Problem 22)AMC 12B 2024 (Problem 20)AMC 12B 2024 (Problem 22)