|
|
| (3 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| − | | + | Problems will be written in 2027-29 |
| − | Work in progress
| |
| − | | |
| − | ==Problem 1==
| |
| − | The diagram below shows the circular face of a clock with radius <math>20</math> cm and a circular disk with radius <math>10</math> cm externally tangent to the clock face at <math>12</math> o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction? [asy]size(170);defaultpen(linewidth(0.9)+fontsize(13pt));draw(unitcircle^^circle((0,1.5),0.5)); path arrow = origin--(-0.13,-0.35)--(-0.06,-0.35)--(-0.06,-0.7)--(0.06,-0.7)--(0.06,-0.35)--(0.13,-0.35)--cycle; for(int i=1;i<=12;i=i+1){draw(0.9*dir(90-30*i)--dir(90-30*i));label("<math>"+(string) i+"</math>",0.78*dir(90-30*i));} dot(origin);draw(shift((0,1.87))*arrow);draw(arc(origin,1.5,68,30),EndArrow(size=12));[/asy]
| |
| − | | |
| − | <math>\textbf{(A) }\text{2 o' clock} \qquad\textbf{(B) }\text{3 o' clock} \qquad\textbf{(C) }\text{4 o' clock} \qquad\textbf{(D) }\text{6 o' clock} \qquad\textbf{(E) }\text{8 o' clock}</math>
| |
| − | | |
| − | ===Problem 2===
| |
| − | | |
| − | Prove that for any positive integer <math>k,</math>
| |
| − | <cmath>\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}</cmath>
| |
| − | is an integer.
| |
| − | | |
| − | ===Problem 3===
| |
| − | (<math>*</math>) Let <math>ABC</math> be a scalene triangle with circumcircle <math>\Omega</math> and incenter <math>I</math>. Ray <math>AI</math> meets <math>\overline{BC}</math> at <math>D</math> and meets <math>\Omega</math> again at <math>M</math>; the circle with diameter <math>\overline{DM}</math> cuts <math>\Omega</math> again at <math>K</math>. Lines <math>MK</math> and <math>BC</math> meet at <math>S</math>, and <math>N</math> is the midpoint of <math>\overline{IS}</math>. The circumcircles of <math>\triangle KID</math> and <math>\triangle MAN</math> intersect at points <math>L_1</math> and <math>L_2</math>. Prove that <math>\Omega</math> passes through the midpoint of either <math>\overline{IL_1}</math> or <math>\overline{IL_2}</math>.
| |
| − | | |
| − | [[2017 USAMO Problems/Problem 3|Solution]]
| |
| − | | |
| − | ===Problem 4===
| |
| − | Find the minimum possible value of <cmath>\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}</cmath>given that <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are nonnegative real numbers such that <math>a+b+c+d=4</math>.
| |
| − | | |
| − | [[2017 USAMO Problems/Problem 4|Solution]]
| |
Latest revision as of 17:41, 4 March 2018
Problems will be written in 2027-29
