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Geometry Math Olympiad PRMO

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

Check the Answer


Answer: is 55.

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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Categories
Geometry Math Olympiad PRMO

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

Check the Answer


Answer: is 51.

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.

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