Difference between revisions of "2015 AIME II Problems/Problem 11"
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===Solution 1=== | ===Solution 1=== | ||
− | Call | + | Call <math>M</math> and <math>N</math> the feet of the altitudes from <math>O</math> to <math>BC</math> and <math>AB</math>, respectively. Let <math>OB = r</math> . Notice that <math>\triangle{OMB} \sim \triangle{QOB}</math> because both are right triangles, and <math>\angle{OBQ} \cong \angle{OBM}</math>. By <math>\frac{MB}{BO}=\frac{BO}{BQ}</math>, <math>MB = r\left(\frac{r}{4.5}\right) = \frac{r^2}{4.5}</math>. However, since <math>O</math> is the circumcenter of triangle <math>ABC</math>, <math>OM</math> is a perpendicular bisector by the definition of a circumcenter. Hence, <math>\frac{r^2}{4.5} = 2 \implies r = 3</math>. Since we know <math>BN=\frac{5}{2}</math> and <math>\triangle BOP \sim \triangle BNO</math>, we have <math>\frac{BP}{3} = \frac{3}{\frac{5}{2}}</math>. Thus, <math>BP = \frac{18}{5}</math>. <math>m + n=\boxed{023}</math>. |
===Solution 2=== | ===Solution 2=== | ||
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Using Power of a Point on <math>Q</math> relative to circle <math>O</math>, we get that <math>x(x+2r) = 0.5 \cdot 4.5 = \frac{9}{4}</math>. Using Pythagorean Theorem on triangle <math>QOB</math> to get <math>(x + r)^2 + r^2 = \frac{81}{4}</math>. Subtracting the first equation from the second, we get that <math>2r^2 = 18</math> and therefore <math>r = 3</math>. Now, set <math>\cos{ABC} = y</math>. Using law of cosines on <math>ABC</math> to find <math>AC</math> in terms of <math>y</math> and plugging that into the extended law of sines, we get <math>\frac{\sqrt{4^2 + 5^2 - 2 \cdot 4 \cdot 5 x}}{\sqrt{1 - x^2}} = 2R = 6</math>. Squaring both sides and cross multiplying, we get <math>36x^2 - 40x + 5 = 0</math>. Now, we get <math>x = \frac{10 \pm \sqrt{55}}{18}</math> using quadratic formula. If you drew a decent diagram, <math>B</math> is acute and therefore <math>x = \frac{10 + \sqrt{55}}{18}</math>(You can also try plugging in both in the end and seeing which gives a rational solution). Note that <math>BP = 3\frac{1}{\sin{OPB}} = \frac{3}{\cos{\angle ABC - \angle QBO}}.</math> Using the cosine addition formula and then plugging in what we know about <math>QBO</math>, we get that <math>BP = \frac{162}{2\cos{B} + \sqrt{5}\sin{B}}</math>. Now, the hard part is to find what <math>\sin{B}</math> is. We therefore want <math>\frac{\sqrt{324 - (10 + \sqrt{55})^2}}{18} = \frac{\sqrt{169 - 20\sqrt{55}}}{18}</math>. For the numerator, by inspection <math>(a + b\sqrt{55})^2</math> will not work for integers <math>a</math> and <math>b</math>. The other case is if there is <math>(a\sqrt{5} + b\sqrt{11})^2</math>. By inspection, <math>5\sqrt{5} - 2\sqrt{11}</math> works. Therefore, plugging all this in yields the answer, <math>\frac{18}{5} \rightarrow \boxed{23}</math>. Solution by hyxue | Using Power of a Point on <math>Q</math> relative to circle <math>O</math>, we get that <math>x(x+2r) = 0.5 \cdot 4.5 = \frac{9}{4}</math>. Using Pythagorean Theorem on triangle <math>QOB</math> to get <math>(x + r)^2 + r^2 = \frac{81}{4}</math>. Subtracting the first equation from the second, we get that <math>2r^2 = 18</math> and therefore <math>r = 3</math>. Now, set <math>\cos{ABC} = y</math>. Using law of cosines on <math>ABC</math> to find <math>AC</math> in terms of <math>y</math> and plugging that into the extended law of sines, we get <math>\frac{\sqrt{4^2 + 5^2 - 2 \cdot 4 \cdot 5 x}}{\sqrt{1 - x^2}} = 2R = 6</math>. Squaring both sides and cross multiplying, we get <math>36x^2 - 40x + 5 = 0</math>. Now, we get <math>x = \frac{10 \pm \sqrt{55}}{18}</math> using quadratic formula. If you drew a decent diagram, <math>B</math> is acute and therefore <math>x = \frac{10 + \sqrt{55}}{18}</math>(You can also try plugging in both in the end and seeing which gives a rational solution). Note that <math>BP = 3\frac{1}{\sin{OPB}} = \frac{3}{\cos{\angle ABC - \angle QBO}}.</math> Using the cosine addition formula and then plugging in what we know about <math>QBO</math>, we get that <math>BP = \frac{162}{2\cos{B} + \sqrt{5}\sin{B}}</math>. Now, the hard part is to find what <math>\sin{B}</math> is. We therefore want <math>\frac{\sqrt{324 - (10 + \sqrt{55})^2}}{18} = \frac{\sqrt{169 - 20\sqrt{55}}}{18}</math>. For the numerator, by inspection <math>(a + b\sqrt{55})^2</math> will not work for integers <math>a</math> and <math>b</math>. The other case is if there is <math>(a\sqrt{5} + b\sqrt{11})^2</math>. By inspection, <math>5\sqrt{5} - 2\sqrt{11}</math> works. Therefore, plugging all this in yields the answer, <math>\frac{18}{5} \rightarrow \boxed{23}</math>. Solution by hyxue | ||
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===Solution 6=== | ===Solution 6=== |
Revision as of 21:33, 21 February 2021
Contents
Problem
The circumcircle of acute has center . The line passing through point perpendicular to intersects lines and and and , respectively. Also , , , and , where and are relatively prime positive integers. Find .
Diagram
Solution
Solution 1
Call and the feet of the altitudes from to and , respectively. Let . Notice that because both are right triangles, and . By , . However, since is the circumcenter of triangle , is a perpendicular bisector by the definition of a circumcenter. Hence, . Since we know and , we have . Thus, . .
Solution 2
Notice that , so . From this we get that . So , plugging in the given values we get , so , and .
Solution 3
Let . Drawing perpendiculars, and . From there, . Thus, . Using , we get . Now let's find . After some calculations with ~ , . Therefore, . .
Solution 4
Let . Extend to touch the circumcircle at a point . Then, note that . But since is a diameter, , implying . It follows that is a cyclic quadrilateral.
Let . By Power of a Point, The answer is .
Solution 5
Note: This is not a very good solution, but it is relatively natural and requires next to no thinking.
Denote the circumradius of to be , the circumcircle of to be , and the shortest distance from to circle to be .
Using Power of a Point on relative to circle , we get that . Using Pythagorean Theorem on triangle to get . Subtracting the first equation from the second, we get that and therefore . Now, set . Using law of cosines on to find in terms of and plugging that into the extended law of sines, we get . Squaring both sides and cross multiplying, we get . Now, we get using quadratic formula. If you drew a decent diagram, is acute and therefore (You can also try plugging in both in the end and seeing which gives a rational solution). Note that Using the cosine addition formula and then plugging in what we know about , we get that . Now, the hard part is to find what is. We therefore want . For the numerator, by inspection will not work for integers and . The other case is if there is . By inspection, works. Therefore, plugging all this in yields the answer, . Solution by hyxue
Solution 6
Reflect , across to points and , respectively with on the circle and collinear. Now, by parallel lines. From here, as collinear. From here, is cyclic, and by power of a point we obtain . ~awang11's sol
See also
2015 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.