Lecture 2 Number Theory --- Miscellaneous Theorems for PRMO, RMO, NMTC, NSEJS, KVPY, AIME, AMC

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Find the sum of all real x such that x^2+x-312+|x^2-x+312|=2(343x-2022). Find the sum of all real x such that x^2+x-312+|x^2-x+312|=2(343x-2022).
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If equation x^2+px+k=0 has equal roots and -4 is a root of quadratic equation x^2+px-4=0, find k. If equation x^2+px+k=0 has equal roots and -4 is a root of quadratic equation x^2+px-4=0, find k.
2-digit number obtained by deleting the middle number is a ninth of original number? Find number 2-digit number obtained by deleting the middle number is a ninth of original number? Find number
The highest power of 200 in 90! is m. The highest power of 9 in 30! is n. What is m+n. The highest power of 200 in 90! is m. The highest power of 9 in 30! is n. What is m+n.
Find the value of cos^2โกใ€–ใ€–10ใ€—^0 ใ€—-cosโกใ€–ใ€–10ใ€—^0 ใ€— cosโกใ€–ใ€–50ใ€—^0 ใ€—+cos^2โกใ€–ใ€–50ใ€—^0 ใ€—. Find the value of cos^2โกใ€–ใ€–10ใ€—^0 ใ€—-cosโกใ€–ใ€–10ใ€—^0 ใ€— cosโกใ€–ใ€–50ใ€—^0 ใ€—+cos^2โกใ€–ใ€–50ใ€—^0 ใ€—.
Find the number of pairs (x,y) of positive integers satisfying the equation โˆšx+โˆšy=โˆš2023. Find the number of pairs (x,y) of positive integers satisfying the equation โˆšx+โˆšy=โˆš2023.