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Find the largest positive integer

This problem is from the 2026 February Harvard-MIT Math Tournament.

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Find all pairs (x, p)
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Find all pairs (x, p)

Find All Prime Pairs (p. q) for Positive Integer n.
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Find All Prime Pairs (p. q) for Positive Integer n.

Find the number of real solutions
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Find the number of real solutions

Find all natural numbers such that 2^n + n divides 8^n + n
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Find all natural numbers such that 2^n + n divides 8^n + n

Find all natural numbers satissfying the equation
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Find all natural numbers satissfying the equation

Solving Lights Out Puzzles | Light Chasing vs Linear Algebra
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Solving Lights Out Puzzles | Light Chasing vs Linear Algebra

a * b = ab + a + b
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a * b = ab + a + b

Find the real solutions
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Find the real solutions

How 𝑒 Was Discovered
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How 𝑒 Was Discovered

Nobody Explained the Schrödinger Equation Like THIS!
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Nobody Explained the Schrödinger Equation Like THIS!

A fun proof for an integer
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A fun proof for an integer

Find all integer k if sqrt(k^2 -pk) is a positive integer for prime p.
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Find all integer k if sqrt(k^2 -pk) is a positive integer for prime p.

the famous goat problem
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the famous goat problem

Compute the largest number
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Compute the largest number

What Lies *Between* a Function and Its Derivative?
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What Lies *Between* a Function and Its Derivative?

Why was this visual proof missed for 400 years? (Fermat's two square theorem)
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Why was this visual proof missed for 400 years? (Fermat's two square theorem)

x^4 - 3x +1 is never a perfect square for positive integer x.
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x^4 - 3x +1 is never a perfect square for positive integer x.

if x+y=8, find the max of x^y (Lambert W function)
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if x+y=8, find the max of x^y (Lambert W function)

Tricks Cops Use the SECOND You Roll Down Your Window
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Tricks Cops Use the SECOND You Roll Down Your Window

Square root of imaginary number.
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Square root of imaginary number.

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