Found problems: 12
2016 BAMO, 3
The ${\textit{distinct prime factors}}$ of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$.
Let $A=2^k - 2$ and $B= 2^k \cdot A$, where $k$ is an integer ($k \ge 2$).
Show that, for every integer $k$ greater than or equal to $2$,
[list=i]
[*] $A$ and $B$ have the same set of distinct prime factors.
[*] $A+1$ and $B+1$ have the same set of distinct prime factors.
[/list]
2016 BAMO, 4
In an acute triangle $ABC$ let $K,L,$ and $M$ be the midpoints of sides $AB,BC,$ and $CA,$ respectively. From each of $K,L,$ and $M$ drop two perpendiculars to the other two sides of the triangle; e.g., drop perpendiculars from $K$ to sides $BC$ and $CA,$ etc. The resulting $6$ perpendiculars intersect at points $Q,S,$ and $T$ as in the figure to form a hexagon $KQLSMT$ inside triangle $ABC.$ Prove that the area of this hexagon $KQLSMT$ is half of the area of the original triangle $ABC.$
[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra; diagram by adihaya*/
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = 11.888712276357234, xmax = 17.841346447833423, ymin = 10.61620970860601, ymax = 15.470685507068502; /* image dimensions */
pen zzttqq = rgb(0.6,0.2,0.); pen qqwuqq = rgb(0.,0.39215686274509803,0.);
pair A = (12.488234161849352,12.833838721895551), B = (16.50823416184936,15.093838721895553), C = (16.28823416184936,11.353838721895551), K = (14.498234161849355,13.963838721895552), L = (16.39823416184936,13.223838721895552), M = (14.388234161849356,12.093838721895551), D = (13.615830174638527,13.467760858438725), F = (15.75135711740064,11.562938202365055), G = (15.625830174638523,14.597760858438724), H = (16.435061748056253,13.849907687412797), T = (14.02296781802369,12.74356027153236), Q = (16.032967818023693,13.873560271532357), O = (16.325061748056253,11.979907687412794);
draw(A--B--C--cycle, zzttqq);
draw((13.426050287639166,13.361068683160477)--(13.532742462917415,13.171288796161116)--(13.722522349916774,13.277980971439364)--D--cycle, qqwuqq);
draw((14.054227993863618,12.223925334689998)--(14.133240861538676,12.426796211152979)--(13.930369985075695,12.505809078828037)--(13.851357117400637,12.302938202365056)--cycle, qqwuqq);
draw((16.337846386707046,12.19724654447628)--(16.12050752964356,12.210031183127075)--(16.107722890992765,11.992692326063588)--O--cycle, qqwuqq);
draw((15.830369985075697,11.765809078828037)--(15.627499108612716,11.844821946503092)--(15.54848624093766,11.641951070040111)--F--cycle, qqwuqq);
draw((15.436050287639164,14.491068683160476)--(15.542742462917412,14.301288796161115)--(15.73252234991677,14.407980971439365)--G--cycle, qqwuqq);
draw((16.217722890992764,13.86269232606359)--(16.20493825234197,13.645353469000101)--(16.42227710940546,13.63256883034931)--H--cycle, qqwuqq);
Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true);
yaxis(ymin, ymax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */
/* draw figures */
draw(A--B, zzttqq);
draw(B--C, zzttqq);
draw(C--A, zzttqq);
draw(M--D);
draw(K--(13.851357117400637,12.302938202365056));
draw(F--L);
draw(L--G);
draw(K--H);
draw(M--O);
/* dots and labels */
dot(A,dotstyle);
label("$A$", (12.52502834296331,12.93568440300881), NE * labelscalefactor);
dot(B,dotstyle);
label("$B$", (16.548187989892043,15.193580123223922), NE * labelscalefactor);
dot(C,dotstyle);
label("$C$", (16.332661580235147,11.457789022504372), NE * labelscalefactor);
dot(K,linewidth(3.pt) + dotstyle);
label("$K$", (14.536608166427676,14.02357961365791), NE * labelscalefactor);
dot(L,linewidth(3.pt) + dotstyle);
label("$L$", (16.43529320388129,13.28463192340569), NE * labelscalefactor);
dot(M,linewidth(3.pt) + dotstyle);
label("$M$", (14.433976542781535,12.155684063298134), NE * labelscalefactor);
dot(D,linewidth(3.pt) + dotstyle);
dot((13.851357117400637,12.302938202365056),linewidth(3.pt) + dotstyle);
dot(F,linewidth(3.pt) + dotstyle);
dot(G,linewidth(3.pt) + dotstyle);
dot(H,linewidth(3.pt) + dotstyle);
dot((15.922967818023695,12.003560271532354),linewidth(3.pt) + dotstyle);
label("$S$", (15.96318773510904,12.063315602016607), NE * labelscalefactor);
dot(T,linewidth(3.pt) + dotstyle);
label("$T$", (14.064502697655428,12.802263292268826), NE * labelscalefactor);
dot(Q,linewidth(3.pt) + dotstyle);
label("$Q$", (16.076082521119794,13.931211152376383), NE * labelscalefactor);
dot(O,linewidth(3.pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]
2015 BAMO, 1
There are $ 7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box #$1$, the second in box #$2$, and so forth up to the seventh card which is placed in box #$7$. You then start working back in the other direction, placing the eighth card in box #$6$, the ninth in box #$5$, up to the thirteenth card being placed in box #$1$. The fourteenth card is then placed in box #$2$, and this continues until every card is distributed. What box will the last card be placed in?
2015 BAMO, 2
Members of a parliament participate in various committees. Each committee consists of at least $2$ people, and it is known that every two committees have at least one member in common. Prove that it is possible to give each member a colored hat (hats are available in black, white or red) so that every committee contains at least $2$ members with different hat colors.
2016 BAMO, 5
The corners of a fixed convex (but not necessarily regular) $n$-gon are labeled with distinct letters. If an observer stands at a point in the plane of the polygon, but outside the polygon, they see the letters in some order from left to right, and they spell a "word" (that is, a string of letters; it doesn't need to be a word in any language). For example, in the diagram below (where $n=4$), an observer at point $X$ would read "$BAMO$," while an observer at point $Y$ would read "$MOAB$."
[center]Diagram to be added soon[/center]
Determine, as a formula in terms of $n$, the maximum number of distinct $n$-letter words which may be read in this manner from a single $n$-gon. Do not count words in which some letter is missing because it is directly behind another letter from the viewer's position.
2014 Contests, 1
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.
2014 BAMO, 1
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.
2014 Contests, 2
There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.)
[center][img]http://i.imgur.com/4ggOa8m.png[/img][/center]
[list=a]
[*]Can the puzzle be solved for $n=5$?
[*]Can the puzzle be solved for $n=2014$?
[/list]
In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.
2015 BAMO, 4
In a quadrilateral, the two segments connecting the midpoints of its opposite sides are equal in length. Prove that the diagonals of the quadrilateral are perpendicular.
(In other words, let $M,N,P,$ and $Q$ be the midpoints of sides $AB,BC,CD,$ and $DA$ in quadrilateral $ABCD$. It is known that segments $MP$ and $NQ$ are equal in length. Prove that $AC$ and $BD$ are perpendicular.)
2014 BAMO, 2
There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.)
[center][img]http://i.imgur.com/4ggOa8m.png[/img][/center]
[list=a]
[*]Can the puzzle be solved for $n=5$?
[*]Can the puzzle be solved for $n=2014$?
[/list]
In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.
2016 BAMO, 1
The diagram below is an example of a ${\textit{rectangle tiled by squares}}$:
[center][img]http://i.imgur.com/XCPQJgk.png[/img][/center]
Each square has been labeled with its side length. The squares fill the rectangle without overlapping. In a similar way, a rectangle can be tiled by nine squares whose side lengths are $2,5,7,9,16,25,28,33$, and $36$. Sketch one such possible arrangement of those squares. They must fill the rectangle without overlapping. Label each square in your sketch by its side length as in the picture above.
2016 BAMO, 2
A weird calculator has a numerical display and only two buttons, $\boxed{D\sharp}$ and $\boxed{D\flat}$. The first button doubles the displayed number and then adds $1$. The second button doubles the displayed number and then subtracts $1$. For example, if the display is showing $5$, then pressing the $\boxed{D\sharp}$ produces $11$. If the display shows $5$ and we press $\boxed{D\flat}$, we get $9$. If the display shows $5$ and we press the sequence $\boxed{D\sharp}$, $\boxed{D\flat}$, $\boxed{D\sharp}$, $\boxed{D\sharp}$, we get a display of $87$.
[list=i]
[*] Suppose the initial displayed number is $1$. Give a sequence of exactly eight button presses that will result in a display of $313$.
[*] Suppose the initial displayed number is $1$, and we then perform exactly eight button presses. Describe all the numbers that can possibly result? Prove your answer by explaining how all these numbers can be produced and that no other numbers can be produced. [/list]