Found problems: 248
1999 USAMTS Problems, 1
The number $N$ consists of $1999$ digits such that if each pair of consecutive digits in $N$ were viewed as a two-digit number, then that number would either be a multiple of $17$ or a multiple of $23$. THe sum of the digits of $N$ is $9599$. Determine the rightmost ten digits of $N$.
1998 USAMTS Problems, 3
Nine cards can be numbered using positive half-integers $(1/2, 1, 3/2, 2, 5/2, \dots )$ so that the sum of the numbers on a randomly chosen pair of cards gives an integer from $2$ to $12$ with the same frequency of occurrence as rolling that sum on two standard dice. What are the numbers on the nine cards and how often does each number appear on the cards?
2005 USAMTS Problems, 5
Lisa and Bart are playing a game. A round table has $n$ lights evenly spaced around its circumference. Some of the lights are on and some of them off; the initial configuration is random. Lisa wins if she can get all of the lights turned on; Bart wins if he can prevent this from happening.
On each turn, Lisa chooses the positions at which to flip the lights, but before the lights are flipped, Bart, knowing Lisa’s choices, can rotate the table to any position that he chooses (or he can leave the table as is). Then the lights in the positions that Lisa chose are flipped: those that are off are turned on and those that are on are turned off.
Here is an example turn for $n = 5$ (a white circle indicates a light that is on, and a black
circle indicates a light that is off):
[asy]
size(250); defaultpen(linewidth(1)); picture p = new picture;
real r = 0.2; pair s1=(0,-4), s2=(0,-8); int[][] filled = {{1,2,3},{1,2,5},{2,3,4,5}};
draw(p,circle((0,0),1));
for(int i = 0; i < 5; ++i) {
pair P = dir(90-72*i); filldraw(p,circle(P,r),white); label(p,string(i+1),P,2*P,fontsize(10));
}
add(p); add(shift(s1)*p); add(shift(s2)*p);
for(int j = 0; j < 3; ++j)
for(int i = 0; i < filled[j].length; ++i)
filldraw(circle(dir(90-72*(filled[j][i]-1))+j*s1,r));
label("$\parbox{15em}{Initial Position.}$", (-4.5,0));
label("$\parbox{15em}{Lisa says ``1,3,4.'' \\ Bart rotates the table one \\ position counterclockwise. }$", (-4.5,0)+s1);
label("$\parbox{15em}{Lights in positions 1,3,4 are \\ flipped.}$", (-4.5,0)+s2);[/asy]
Lisa can take as many turns as she needs to win, or she can give up if it becomes clear
to her that Bart can prevent her from winning.
(a) Show that if $n = 7$ and initially at least one light is on and at least one light is off,
then Bart can always prevent Lisa from winning.
(b) Show that if $n = 8$, then Lisa can always win in at most 8 turns.
2020 USAMTS Problems, 4:
Two beasts, Rosencrans and Gildenstern, play a game. They have a circle with $n$ points ($n \ge 5$) on it. On their turn, each beast (starting with Rosencrans) draws a chord between a pair of points in such a way that any two chords have a shared point. (The chords either intersect or have a common endpoint.) For example, two potential legal moves for the second player are drawn below with dotted lines.
[asy]
unitsize(0.7cm);
draw(circle((0,0),1));
dot((0,-1));
pair A = (-1/2,-(sqrt(3))/2);
dot(A);
pair B = ((sqrt(2))/2,-(sqrt(2))/2);
dot(B);
pair C = ((sqrt(3))/2,1/2);
dot(C);
draw(A--C);
pair D = (-(sqrt(0.05)),sqrt(0.95));
dot(D);
pair E = (-(sqrt(0.2)),sqrt(0.8));
dot(E);
draw(B--E,dotted);
draw(C--D,dotted);
[/asy]
The game ends when a player cannot draw a chord. The last beast to draw a chord wins. For which $n$ does Rosencrans win?
2020 USAMTS Problems, 4:
In a group of $n > 20$ people, there are some (at least one, and possibly all) pairs of people that know each other. Knowing is symmetric; if Alice knows Blaine, then Blaine also knows Alice. For some values of $n$ and $k,$ this group has a peculiar property: If any $20$ people are removed from the group, the number of pairs of people that know each other is
at most $\frac{n-k}{n}$ times that of the original group of people.
(a) If $k = 41,$ for what positive integers $n$ could such a group exist?
(b) If $k = 39,$ for what positive integers $n$ could such a group exist?
2016 USAMTS Problems, 4:
Find all functions $f(x)$ from nonnegative reals to nonnegative reals such that $f(f(x))=x^4$ and $f(x)\leq Cx^2$ for some constant $C$.
2005 USAMTS Problems, 3
We play a game. The pot starts at $\$0$. On every turn, you flip a fair coin. If you flip heads, I add $\$100$ to the pot. If you flip tails, I take all of the money out of the pot, and you are assessed a "strike". You can stop the game before any flip and collect the contents of the pot, but if you get 3 strikes, the game is over and you win nothing. Find, with proof, the expected value of your winnings if you follow an optimal strategy.
2009 USAMTS Problems, 3
A square of side length $5$ is inscribed in a square of side length $7$. If we construct a grid of $1\times1$ squares for both squares, as shown to the right, then we fi nd that the two grids have $8$ lattice points in common. If we do the same construction by inscribing a square of side length $1489$ in a square of side length $2009$, and construct a grid of $1\times1$ squares in each large square, then how many lattice points will the two grids of $1\times1$ squares have in common?
[asy]
import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=11.88,ymin=-4.69,ymax=8.77;
pair H_2=(0,3), I_2=(3,7), J_2=(7,4), K_2=(4,0), L_2=(3.01,1.99), M_2=(2.01,4), N_2=(4.01,5.01), O_2=(5.01,3);
draw((0,0)--(0,7)); draw((0,7)--(7,7)); draw((7,7)--(7,0)); draw((7,0)--(0,0)); draw((0,6)--(7,6)); draw((0,5)--(7,5)); draw(J_2--(0,4)); draw(H_2--(7,3)); draw((0,2)--(7,2)); draw((0,1)--(7,1)); draw((1,0)--(1,7)); draw((2,7)--(2,0)); draw((3,0)--I_2); draw(K_2--(4,7)); draw((5,0)--(5,7)); draw((6,7)--(6,0)); draw(H_2--I_2); draw(I_2--J_2); draw(J_2--K_2); draw(K_2--H_2); draw(H_2--I_2); draw(I_2--J_2); draw((2.41,6.21)--(6.4,3.2)); draw((5.8,2.4)--(1.81,5.41)); draw((1.2,4.61)--(5.2,1.6)); draw((4.6,0.8)--(0.6,3.8)); draw((3.8,6.4)--(0.8,2.4)); draw((1.61,1.79)--(4.6,5.8)); draw((5.4,5.2)--(2.41,1.19)); draw((3.21,0.59)--(6.2,4.6)); draw((0,7)--(7,7),linewidth(1.2)); draw((7,7)--(7,0),linewidth(1.2)); draw((0,0)--(7,0),linewidth(1.2)); draw((0,7)--(0,0),linewidth(1.2));
dot(H_2,linewidth(4pt)+ds); dot(I_2,linewidth(4pt)+ds); dot(J_2,linewidth(4pt)+ds); dot(K_2,linewidth(4pt)+ds); dot(L_2,linewidth(4pt)+ds); dot(M_2,linewidth(4pt)+ds); dot(N_2,linewidth(4pt)+ds); dot(O_2,linewidth(4pt)+ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]
2015 AMC 10, 22
In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$?
[asy]
import cse5;pathpen=black;pointpen=black;
size(2inch);
pair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162);
D(MP("A",A,A)--MP("B",B,B)--MP("C",C,C)--MP("D",D,D)--MP("E",E,E)--cycle,linewidth(1.5));
D(A--C--E--B--D--cycle);
pair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A);
D(MP("F",F,dir(126))--MP("I",I,dir(270))--MP("G",G,dir(54))--MP("J",J,dir(198))--MP("H",H,dir(342))--cycle);
[/asy]
$\textbf{(A) } 3
\qquad\textbf{(B) } 12-4\sqrt5
\qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3}
\qquad\textbf{(D) } 1+\sqrt5
\qquad\textbf{(E) } \dfrac{11+11\sqrt5}{10}
$
2006 AIME Problems, 5
The number \[ \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006} \] can be written as $a\sqrt{2}+b\sqrt{3}+c\sqrt{5},$ where $a, b,$ and $c$ are positive integers. Find $a\cdot b\cdot c$.
2004 USAMTS Problems, 5
Medians $AD$, $BE$, and $CF$ of triangle $ABC$ meet at $G$ as shown. Six small triangles, each with vertex at $G$, are formed. We draw the circles inscribed in triangles $AFG$, $BDG$, and $CDG$ as shown. Prove that if these three circles are all congruent, then $ABC$ is equilateral.
[asy]
size(200);
defaultpen(fontsize(10));
pair C=origin, B=(12,0), A=(3,14), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);
draw(A--B--C--A--D^^B--E^^C--F);
draw(incircle(C,G,D)^^incircle(G,D,B)^^incircle(A,F,G));
pair point=G;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$G$", G, dir(7));[/asy]
2019 USAMTS Problems, 2
2/1/31. Let $x, y,$ and $z$ be real numbers greater than $1$. Prove that if $x^y = y^z = z^x$, then $x = y = z$.
2009 USAMTS Problems, 5
The sequences $(a_n), (b_n),$ and $(c_n)$ are de fined by $a_0 = 1, b_0 = 0, c_0 = 0,$ and
\[a_n = a_{n-1} +
\frac{c_{n-1}}{n}, b_n = b_{n-1} +\frac{a_{n-1}}{n}, c_n = c_{n-1} +\frac{b_{n-1}}{n}\]
for all $n \geq1$. Prove that
\[\left|a_n -\frac{n + 1}{3}\right|<\frac{2}{\sqrt{3n}}\]
for all $n \geq 1$.
1999 USAMTS Problems, 1
Define the unique $9$-digit integer $M$ that has the following properties: (1) its digits are all distinct and nonzero; and (2) for every positive integer $m=2,3,4,...,9$, the integer formed by the leftmost $m$ digits of $M$ is divisible by $m$.
2011 USAMTS Problems, 2
Four siblings are sitting down to eat some mashed potatoes for lunch: Ethan has 1 ounce of mashed potatoes, Macey has 2 ounces, Liana has 4 ounces, and Samuel has 8 ounces. This is not fair. A blend consists of choosing any two children at random, combining their plates of mashed potatoes, and then giving each of those two children half of the combination. After the children's father performs four blends consecutively, what is the probability that the four children will all have the same amount of mashed potatoes?
2017 USAMTS Problems, 2
After each Goober ride, the driver rates the passenger as $1$, $2$, $3$, $4$, or $5$ stars. The passenger's overall rating is determined as the average of all of the ratings given to him or her by drivers so far. Noah had been on several rides, and his rating was neither $1$ nor $5$. Then he got a $1$ star on a ride because he barfed on the driver. Show that the number of $5$ stars that Noah needs in order to climb back to at least his overall rating before bar ng is independent of the number of rides that he had taken.
2012 USAMTS Problems, 5
A unit square $ABCD$ is given in the plane, with $O$ being the intersection of its diagonals. A ray $l$ is drawn from $O$. Let $X$ be the unique point on $l$ such that $AX + CX = 2$, and let $Y$ be the point on $l$ such that $BY + DY = 2$. Let $Z$ be the midpoint of $\overline{XY}$, with $Z = X$ if $X$ and $Y$ coincide. Find, with proof, the minimum value of the length of $OZ$.
2019 USAMTS Problems, 2
A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or
lime with equal probability and gives the grid to her friend Sophia.
Sophia then plays with the grid of blocks. She can take the top row of blocks and move
it to the bottom, as shown.
1 2 3
4 5 6
7 8 9
4 5 6
7 8 9
1 2 3
Grid A Grid A0
She can also take the leftmost column of blocks and move it to the right end, as shown.
1 2 3
4 5 6
7 8 9
2 3 1
5 6 4
8 9 7
Grid B Grid B0
Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the
moves described above to obtain another grid with the same coloring but a different numbering
scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the
top row of blocks to the bottom results in a grid with a different numbering but the same
coloring as Grid A.
What is the probability that Sophia receives a citrus grid of blocks?
1999 USAMTS Problems, 3
Determine the value of \[S=\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\cdots+\sqrt{1+\dfrac{1}{1999^2}+\dfrac{1}{2000^2}}\]
2012 USAMTS Problems, 5
Let $P$ and $Q$ be two polynomials with real coeficients such that $P$ has degree greater than $1$ and \[P(Q(x)) = P(P(x)) + P(x).\]Show that $P(-x) = P(x) + x$.
2005 USAMTS Problems, 5
Given triangle $ABC$, let $M$ be the midpoint of side $AB$ and $N$ be the midpoint of side $AC$. A circle is inscribed inside quadrilateral $NMBC$, tangent to all four sides, and that circle touches $MN$ at point $X.$ The circle inscribed in triangle $AMN$ touches $MN$ at point $Y$, with $Y$ between $X$ and $N$. If $XY=1$ and $BC=12$, find, with proof, the lengths of the sides $AB$ and $AC$.
2019 USAMTS Problems, 2
An apple orchard’s layout is a rectangular grid of unit squares. Some pairs of adjacent squares have a thick wall of grape vines between them. The orchard wants to post some robot sentries to guard its prized apple trees. Each sentry occupies a single square of the layout, and from there it can guard both its square and any square in the same row and column that it can see, where only walls and the edges of the orchard block its sight. A sample layout (not the layout of the actual orchard, which is not given) is shown below.
Although a square may be guarded by multiple sentries, the sentries have not been programmed to avoid attacking other sentries. Thus, no sentry may be placed on a square guarded by another sentry. The orchard’s expert has found a way to guard all the squares of the orchard by placing 1000 sentries. However, the contractor shipped 2020 sentries. Show that it is impossible for the orchard to place all 2020 of the sentries without two of them attacking each other.
2007 AIME Problems, 9
In right triangle $ABC$ with right angle $C$, $CA=30$ and $CB=16$. Its legs $\overline{CA}$ and $\overline{CB}$ are extended beyond $A$ and $B$. Points $O_{1}$ and $O_{2}$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_{1}$ is tangent to the hypotenuse and to the extension of leg CA, the circle with center $O_{2}$ is tangent to the hypotenuse and to the extension of leg CB, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2008 ITest, 75
Let \[S=\sqrt{1+\dfrac1{1^2}+\dfrac1{2^2}}+\sqrt{1+\dfrac1{2^2}+\dfrac1{3^2}}+\cdots+\sqrt{1+\dfrac1{2007^2}+\dfrac1{2008^2}}.\] Compute $\lfloor S^2\rfloor$.
2019 USAMTS Problems, 3
A positive integer $n > 1$ is juicy if its divisors $d_1 < d_2 < \dots < d_k$ satisfy $d_i - d_{i-1} \mid n$ for all $2 \leq i \leq k$. Find all squarefree juicy integers.