Found problems: 248
2009 USAMTS Problems, 3
We are given a rectangular piece of white paper with length $25$ and width $20$. On the paper we color blue the interiors of $120$ disjoint squares of side length $1$ (the sides of the squares do not necessarily have to be parallel to the sides of the paper). Prove that we can draw a circle of diameter $1$ on the remaining paper such that the entire interior of the circle is white.
2024 USAMTS Problems, 4
During a lecture, each of $26$ mathematicians falls asleep exactly once, and stays asleep for a nonzero amount of time. Each mathematician is awake at the moment the lecture starts, and the moment the lecture finishes. Prove that there are either $6$ mathematicians such that no two are asleep at the same time, or $6$ mathematicians such that there is some point in time during which all $6$ are asleep.
2004 USAMTS Problems, 3
Define the recursive sequence $1, 4, 13, \ldots$ by $s_1 = 1$ and $s_{n+1} = 3s_n + 1$ for all positive integers $n$. The element $s_{18} = 193710244$ ends in two identical digits. Prove that all the elements in the sequence that end in two or more identical digits come in groups of three consecutive elements that have the same number of identical digits at the end.
2017 USAMTS Problems, 1
Fill each white square in with a number so that each of the $27$ three-digit numbers whose digits are all $1$, $2$, or $3$ is used exactly once. For each pair of white squares sharing a side, the two numbers must have equal digits in exactly two of the three positions (ones, tens, hundreds). Some numbers have been given to you.
You do not need to prove that your answer is the only one possible; you merely need to nd an answer that satis es the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justi cation acceptable.)
[asy]
unitsize(16);
int[][] a = {
{999, 999, 999, 000, 000, 212, 000, 000},
{000, 888, 000, 213, 888, 000, 888, 123},
{000, 888, 000, 000, 000, 000, 131, 000},
{000, 888, 121, 888, 000, 113, 888, 000},
{000, 000, 000, 000, 312, 999, 999, 999}};
for (int i = 0; i < 8; ++i) {
for (int j = 0; j < 5; ++j) {
if (a[j][i] != 999) draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle);
if (a[j][i] == 888) fill((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle);
if (a[j][i] > 0 && a[j][i] < 999) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(8pt));
}
}
[/asy]
[img]https://cdn.artofproblemsolving.com/attachments/f/b/bd9d0902922cd34e6e1b089373e515df698a9f.png[/img]
2013 USAMTS Problems, 1
In the grid shown, fill in each empty space with a number, such that after the grid is completely filled in, the number in each space is equal to the smallest positive integer that does not appear in any of the touching spaces. (A pair of spaces is considered to touch if they both share a vertex.)
You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
unitsize(.5cm);
int count=0;
real s3=1.73205081;
pair[] points={(-3,3*s3), (-1,3*s3), (1,3*s3), (3,3*s3), (-4,2*s3), (-2,2*s3), (0,2*s3), (2,2*s3), (4,2*s3), (-5,s3), (-3,s3), (-1,s3), (1,s3), (3,s3), (5,s3), (-6,0), (-4,0), (-2,0), (0,0), (2,0), (4,0), (6,0), (-5,-s3), (-3,-s3), (-1,-s3), (1,-s3), (3,-s3), (5,-s3), (-4,-2*s3), (-2,-2*s3), (0,-2*s3), (2,-2*s3), (4,-2*s3), (-3,-3*s3), (-1,-3*s3), (1,-3*s3), (3,-3*s3)};
void draw_hexagon(pair p)
{
draw(shift(p)*scale(2/s3)*(dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--dir(30)));
}
void add_number()
{
draw_hexagon(points[count]);
count+=1;
}
void add_number(int n)
{
label((string)n, points[count]);
add_number();
}
add_number(4);
add_number();
add_number();
add_number(1);
add_number(1);
add_number();
add_number(1);
add_number();
add_number(3);
add_number();
add_number(2);
add_number();
add_number();
add_number(6);
add_number();
add_number();
add_number();
add_number();
add_number(7);
add_number();
add_number();
add_number();
add_number();
add_number(3);
add_number();
add_number();
add_number(5);
add_number();
add_number(2);
add_number();
add_number(4);
add_number();
add_number(3);
add_number(3);
add_number();
add_number();
add_number(2);[/asy]
2019 USAMTS Problems, 5
Let $ABC$ be a triangle with circumcenter $O$, $A$-excenter $I_A$, $B$-excenter $I_B$, and $C$-excenter $I_C$. The incircle of $\Delta ABC$ is tangent to sides $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. Lines $I_BE$ and $I_CF$ intersect at $P$. If the line through $O$ perpendicular to $OP$ passes through $I_A$, prove that $\angle A = 60^\circ$.
[i]An excenter is the point of concurrency among one internal angle bisector and two external angle bisectors of a triangle.[/i]
2004 USAMTS Problems, 4
How many quadrilaterals in the plane have four of the nine points $(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)$ as vertices? Do count both concave and convex quadrilaterals, but do not count figures where two sides cross each other or where a vertex angle is $180^{\circ}$. Rigorously verify that no quadrilateral was skipped or counted more than once.
[asy]size(50);
dot((0,0));
dot((0,1));
dot((0,2));
dot((1,0));
dot((1,1));
dot((1,2));
dot((2,0));
dot((2,1));
dot((2,2));[/asy]
2013 USAMTS Problems, 3
For each positive integer $n\ge2$, find a polynomial $P_n(x)$ with rational coefficients such that $\displaystyle P_n(\sqrt[n]2)=\frac1{1+\sqrt[n]2}$. (Note that $\sqrt[n]2$ denotes the positive $n^\text{th}$ root of $2$.)
1998 USAMTS Problems, 2
There are infinitely many ordered pairs $(m,n)$ of positive integers for which the sum
\[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\]
is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs.
2002 USAMTS Problems, 2
Given positive integers $p$, $u$, and $v$ such that $u^2+2v^2=p$, determine, in terms of $u$ and $v$, integers $m$ and $n$ such that $3m^2-2mn+3n^2=24p$. (It is known that if $p$ is any prime number congruent to 1 or 3 modulo 8, then we can find integers $u$ and $v$ such that $u^2+2v^2=p$)
2009 USAMTS Problems, 1
Let $ABCD$ be a convex quadrilateral with $AC \perp BD$, and let $P$ be the intersection of $AC$ and $BD$. Suppose that the distance from $P$ to $AB$ is $99$, the distance from $P$ to $BC$ is $63$, and the distance from $P$ to $CD$ is $77$. What is the distance from $P$ to $AD$?
2024 USAMTS Problems, 2
A regular hexagon is placed on top of a unit circle such that one vertex coincides with the center of the circle, exactly two vertices lie on the circumference of the circle, and exactly one vertex lies outside of the circle. Determine the area of the hexagon.
2020 USAMTS Problems, 5:
[b]5/1/32.[/b] Find all pairs of rational numbers $(a, b)$ such that $0 < a < b$ and $a^a = b^b$.
1998 USAMTS Problems, 5
The figure on the right shows the ellipse $\frac{(x-19)^2}{19}+\frac{(x-98)^2}{98}=1998$.
Let $R_1,R_2,R_3,$ and $R_4$ denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of $R_1-R_2+R_3-R_4$.
[asy]
defaultpen(linewidth(0.7));
pair c=(19,98);
real dist = 30;
real a = sqrt(1998*19),b=sqrt(1998*98);
xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow);
yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow);
draw(ellipse(c,a,b));
label("$R_1$",(100,200));
label("$R_2$",(-80,200));
label("$R_3$",(-60,-150));
label("$R_4$",(70,-150));[/asy]
2013 USAMTS Problems, 1
Alex is trying to open a lock whose code is a sequence that is three letters long, with each of the letters being one of $\text A$, $\text B$ or $\text C$, possibly repeated. The lock has three buttons, labeled $\text A$, $\text B$ and $\text C$. When the most recent $3$ button-presses form the code, the lock opens. What is the minimum number of total button presses Alex needs to guarantee opening the lock?
2009 USAMTS Problems, 5
The cubic equation $x^3+2x-1=0$ has exactly one real root $r$. Note that $0.4<r<0.5$.
(a) Find, with proof, an increasing sequence of positive integers $a_1 < a_2 < a_3 < \cdots$ such that
\[\frac{1}{2}=r^{a_1}+r^{a_2}+r^{a_3}+\cdots.\]
(b) Prove that the sequence that you found in part (a) is the unique increasing sequence with the above property.
1982 AMC 12/AHSME, 18
In the adjoining figure of a rectangular solid, $\angle DHG=45^\circ$ and $\angle FHB=60^\circ$. Find the cosine of $\angle BHD$.
[asy]
size(200);
import three;defaultpen(linewidth(0.7)+fontsize(10));
currentprojection=orthographic(1/3+1/10,1-1/10,1/3);
real r=sqrt(3);
triple A=(0,0,r), B=(0,r,r), C=(1,r,r), D=(1,0,r), E=O, F=(0,r,0), G=(1,0,0), H=(1,r,0);
draw(D--G--H--D--A--B--C--D--B--F--H--B^^C--H);
draw(A--E^^G--E^^F--E, linetype("4 4"));
label("$A$", A, N);
label("$B$", B, dir(0));
label("$C$", C, N);
label("$D$", D, W);
label("$E$", E, NW);
label("$F$", F, S);
label("$G$", G, W);
label("$H$", H, S);
triple H45=(1,r-0.15,0.1), H60=(1-0.05, r, 0.07);
label("$45^\circ$", H45, dir(125), fontsize(8));
label("$60^\circ$", H60, dir(25), fontsize(8));[/asy]
$\textbf {(A) } \frac{\sqrt{3}}{6} \qquad \textbf {(B) } \frac{\sqrt{2}}{6} \qquad \textbf {(C) } \frac{\sqrt{6}}{3} \qquad \textbf {(D) } \frac{\sqrt{6}}{4} \qquad \textbf {(E) } \frac{\sqrt{6}-\sqrt{2}}{4}$
2016 USAMTS Problems, 5:
Let $n\geq 4$ and $y_1,\dots, y_n$ real with
$$\sum_{k=1}^n y_k=\sum_{k=1}^n k y_k=\sum_{k=1}^n k^2y_k=0$$
and
$$y_{k+3}-3y_{k+2}+3y_{k+1}-y_k=0$$
for $1\leq k\leq n-3$. Prove that
$$\sum_{k=1}^n k^3y_k=0$$
1999 USAMTS Problems, 2
Let $C$ be the set of non-negative integers which can be expressed as $1999s+2000t$, where $s$ and $t$ are also non-negative integers.
(a) Show that $3,994,001$ is not in $C$.
(b) Show that if $0\leq n \leq 3,994,001$ and $n$ is an integer not in $C$, then $3,994,001-n$ is in $C$.
2011 USAMTS Problems, 5
Let $k>2$ be a positive integer. Elise and Xavier play a game that has four steps, in this order.
[list=1]
[*]Elise picks $2$ nonzero digits $(1-9)$, called $e$ and $f$.
[*]Xavier then picks $k$ nonzero digits $(1-9)$, called $x_1,\cdots,x_k$.
[*]Elise picks any positive integer $d$.
[*]Xaiver picks an integer $b>10$.[/list]
Each player's choices are known to the other player when the choices are made.
The winner is determined as follows. Elise writes down the two-digit base $b$ number $ef_b$. Next, Xavier writes the $k$-digit base $b$ number that is constructed by concatenating his digits,
\[(x_1\cdots x_k)_b.\]
They then compute the greatest common divisor (gcd) of these two numbers. If this gcd is greater than or equal to the integer $d$ then Xavier wins. Otherwise Elise wins.
(As an example game for $k=3$, Elise chooses the digits $(e, f) = (2, 4)$, Xavier chooses $(4, 4, 8)$, and then Elise picks $d = 100$. Xavier picks base $b = 25$. The base-25 numbers $2425$ and $44825$ are, respectively, equal to $54$ and $2608$. The greatest common divisor of these two is $2$, which is much less than $100$, so Elise wins handily.)
Find all $k$ for which Xavier can force a win, no matter how Elise plays.
2004 USAMTS Problems, 2
Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle's area is numerically equal to $6$ times its perimeter.
2004 AMC 10, 16
The $ 5\times 5$ grid shown contains a collection of squares with sizes from $ 1\times 1$ to $ 5\times 5$. How many of these squares contain the black center square?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt));
for(int i=0; i<=5; ++i)
{
draw((0,i)--(5,i));
draw((i,0)--(i,5));
}
fill((2,2)--(2,3)--(3,3)--(3,2)--cycle);[/asy]$ \textbf{(A)}\ 12\qquad
\textbf{(B)}\ 15\qquad
\textbf{(C)}\ 17\qquad
\textbf{(D)}\ 19\qquad
\textbf{(E)}\ 20$
2020 USAMTS Problems, 2:
Infinitely many math beasts stand in a line, all six feet apart, wearing masks, and with clean hands. Grogg starts at the front of the line, holding $n$ pieces of candy, $ n \ge 1,$ and everyone else has none. He passes his candy to the beasts behind him, one piece each to the next $n$ beasts in line. Then, Grogg leaves the line. The other beasts repeat this process: the beast in front, who has $k$ pieces of candy, passes one piece each to the next $k$ beasts in line, and then leaves the line. For some values of $n,$ another beast, besides Grogg, temporarily holds all the candy. For which values of $n$ does this occur?
1998 USAMTS Problems, 3
It is possible to arrange eight of the nine numbers $2, 3, 4, 7, 10, 11, 12, 13, 15$ in the vacant squares of the $3$ by $4$ array shown on the right so that the arithmetic average of the numbers in each row and in each column is the same integer. Exhibit such an arrangement, and specify which one of the nine numbers must be left out when completing the array.
[asy]
defaultpen(linewidth(0.7));
for(int x=0;x<=4;++x)
draw((x+.5,.5)--(x+.5,3.5));
for(int x=0;x<=3;++x)
draw((.5,x+.5)--(4.5,x+.5));
label("$1$",(1,3));
label("$9$",(2,2));
label("$14$",(3,1));
label("$5$",(4,2));[/asy]
2019 USAMTS Problems, 5
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for any reals $x, y,$ $$f(x + y)f(x - y) = (f(x))^2 - (f(y))^2$$. Additionally, suppose that $f(x + 2 \pi) = f(x)$ and that there does not exist a positive real $a < 2 \pi$ such that $f(x + a) = f(x)$ for all reals $x$. Show that for all reals $x$, $$ |f(\frac{\pi}{2})| \geq f(x)$$.