NMTC Stage 2 BHASKARA Junior Level 2025

$$m > 0, n > 0$$ are integers. Find all pairs $$(m, n)$$ such that $$m^2 + 3n$$ and $$n^2 + 3m$$ are both perfect squares simultaneously.

$$PQR$$ is a right triangle with $$\angle QPR = 90^\circ$$. $$I$$ is the incenter of $$\triangle PQR$$. The incircle touches side $$PQ$$ at $$M$$ and side $$PR$$ at $$N$$. A line is drawn through $$I$$ to cut $$PQ$$ at $$A$$ and $$PR$$ at $$B$$. Prove that $$PA \cdot PB \ge 4r^2$$, where $$r$$ is the inradius of $$\triangle PQR$$.

Solve for $$x, y, z$$ from the following simultaneous equations. 

$$x^2 + xy + xz - x = 7$$ 

$$y^2 + yz + yx - y = -14$$

$$z^2 + zx + zy - z = 49$$

Twenty-eight integers are chosen from the interval $$[104, 208]$$. Prove that there exist two of them with a common prime divisor.

In the parallelogram $$ABCD$$, the diagonal $$AC$$ is perpendicular to the side $$AD$$. $$AH$$ is drawn perpendicular to $$CD$$. The tangent at $$D$$ to the circumcircle of $$\triangle ADB$$ meets $$CA$$ produced at $$P$$. Prove that $$\angle PBA = \angle DBH$$.

There are 6 friends, Manish, Cathy, Sundar, Elisha, Lakshaman and Divya. Manish told that the sum of our ages put together is five times my age. He again told that when Sundar will be three times my present age, the sum of my age and Divya's age will be equal to the sum of the present ages of the five of us; Elisha's age will be three times her present age and Lakshaman's age will be twice Sundar's present age plus one year. They all started working to find their present ages and Cathy asked Manish that could he tell something more. Then Manish said that his age is an odd number. How old are Manish and Sundar?

Solve in positive integers $$\frac{x^2-y}{34x-y^2} = \frac{y}{x}$$, provided $$34x - y^2 \ne 0$$.

Let $$ABC$$ be an acute-angled triangle. Let points $$O, H$$ be its circumcenter and orthocenter respectively. The altitude from $$A$$, the perpendicular bisector of side $$AC$$ and side $$BC$$ are concurrent. Then find $$CH \colon BO$$.
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