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Triangles and Internal bisectors | PRMO 2019 | Question 10

Try this beautiful problem from the PRMO, 2019 based on triangles and internal bisectors.

Triangles and internal bisectors – PRMO 2019

Let ABC be a triangle and let D be its circumcircle, The internal bisectors of angles A,B and C intersect D at $A_1,B_1 and C_1$ the internal bisectors of $A_1,B_1,C_1$ of the triangle $A_1B_1C_1$ intersect D at $A_2,B_2,C_2$. If the smallest angle of triangle ABC is 40 find the magnitude of the smallest angle of triangle $A_2B_2C_2$ in degrees.

• is 107
• is 55
• is 840
• cannot be determined from the given information

Key Concepts

Lines

Algebra

Angles

PRMO, 2019, Question 10

Geometry Vol I to IV by Hall and Stevens

Try with Hints

First hint

angle $A_1B_1C_1=90 – \frac{ABC}{2}$ angle $A_1C_1B_1=90-\frac{ACB}{2}$

Second Hint

angle $B_1A_1C_1$=90-$\frac{BAC}{2}$

Final Step

then angle $A_2B_2C_2=90-\frac{90-\frac{ABC}{2}}{2}$=45+$\frac{ABC}{4}$=55.

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Lines and Angles | PRMO 2019 | Question 7

Try this beautiful problem from the PRMO, 2019 based on Lines and Angles.

Lines and Angles – PRMO 2019

On a clock, there are two instants between 12 noon and 1 pm when the hour hand and the minute hand are at right angles. the difference in minutes between these two instants written as a+$\frac{b}{c}$ where a b c are positive integers with $b\lt c$ and $\frac{b}{c}$ in the reduced form. find a+b+c.

• is 107
• is 51
• is 840
• cannot be determined from the given information

Key Concepts

Lines

Algebra

Angles

PRMO, 2019, Question 7

Higher Algebra by Hall and Knight

Try with Hints

First hint

Minute hand turns 6 (in degrees) in 1 min and hour hand turns half (in degree) in 1 min

Second Hint

x min after 12 is 6x -$\frac{x}{2}$=90 or 270 (in degrees) then x =16$\frac{4}{11}$ or 49$\frac{1}{11}$

Final Step

Difference 32$\frac{8}{11}$ then a+b+c=32+8+11=51.