Found problems: 248
2009 USAMTS Problems, 4
The Rational Unit Jumping Frog starts at $(0, 0)$ on the Cartesian plane, and each minute jumps a distance of exactly $1$ unit to a point with rational coordinates.
(a) Show that it is possible for the frog to reach the point $\left(\frac15,\frac{1}{17}\right)$ in a finite amount of time.
(b) Show that the frog can never reach the point $\left(0,\frac14\right)$.
2014 USAMTS Problems, 1:
Divide the grid shown to the right into more than one region so that the following rules are satisfied.
1. Each unit square lies entirely within exactly 1 region.
2. Each region is a single piece connected by the edges of its unit squares.
3. Each region contains the same number of whole unit squares.
4. Each region contains the same sum of numbers.
You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that works. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
size(6cm);
for (int i=0; i<=8; ++i)
draw((i,0)--(i,7), linewidth(.8));
for (int j=0; j<=7; ++j)
draw((0,j)--(8,j), linewidth(.8));
void draw_num(pair ll_corner, int num)
{
label(string(num), ll_corner + (0.5, 0.5), p = fontsize(18pt));
}
draw_num((0, 0), 1);
draw_num((1, 0), 1);
draw_num((2, 0), 1);
draw_num((0, 5), 4);
draw_num((1, 1), 4);
draw_num((1, 4), 3);
draw_num((2, 2), 4);
draw_num((3, 4), 3);
draw_num((3, 5), 2);
draw_num((4, 1), 4);
draw_num((4, 3), 4);
draw_num((5, 4), 4);
draw_num((5, 6), 6);
draw_num((6, 2), 3);
draw_num((6, 5), 4);
draw_num((6, 6), 5);
draw_num((7, 1), 4);
draw_num((7, 6), 6);[/asy]
2015 USAMTS Problems, 2
[b]2/1/27.[/b] Suppose $a, b,$ and $c$ are distinct positive real numbers such that \begin{align*}abc=1000, \\ bc(1-a)+a(b+c)=110.\end{align*}
If $a<1$, show that $10<c<100$.
2020 USAMTS Problems, 5:
Let $a_1$ be any positive integer. For all $i$, write $5^{2020}$ times $a_i$ in base $10$, replace each digit with its remainder when divided by $2$, read off the result in binary, and call that $a_{i+1}$. Prove that $a_N = a_{N+2^{2020}}$ for all sufficiently large $N$.
2010 Contests, 2
There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
2015 USAMTS Problems, 4
Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with
every possible height-weight combination.
The USAMTS wants to field a competitive team, so there are some strict requirements.
[list]
[*] If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
[*] If person $P$ is on the team, then no one whose weight is the same as $P$’s height can also be on the team.
[/list]
Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?
2011 USAMTS Problems, 2
Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$. Compute the sum \[\dfrac{x^2}{x-1}+\dfrac{x^4}{x^2-1}+\dfrac{x^6}{x^3-1}+\cdots+\dfrac{x^{4020}}{x^{2010}-1}.\]
1998 USAMTS Problems, 2
Prove that there are infinitely many ordered triples of positive integers $(a,b,c)$ such that the greatest common divisor of $a,b,$ and $c$ is $1$, and the sum $a^2b^2+b^2c^2+c^2a^2$ is the square of an integer.
2001 Putnam, 2
Find all pairs of real numbers $(x,y)$ satisfying the system of equations:
\begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\
\frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}
2015 USAMTS Problems, 5
Find all positive integers $n$ that have distinct positive divisors $d_1, d_2, \dots, d_k$, where $k>1$, that are in arithmetic progression and $$n=d_1+d_2+\cdots+d_k.$$ Note that $d_1, d_2, \dots, d_k$ do not have to be all the divisors of $n$.
2012 USAMTS Problems, 3
Let $f(x) = x-\tfrac1{x}$, and defi ne $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.
2004 AIME Problems, 12
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6$, $BC=5=DA$, and $CD=4$. Draw circles of radius 3 centered at $A$ and $B$, and circles of radius 2 centered at $C$ and $D$. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p$, where $k$, $m$, $n$, and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p$.
2009 USAMTS Problems, 5
Let $ABC$ be a triangle with $AB = 3, AC = 4,$ and $BC = 5$, let $P$ be a point on $BC$, and let $Q$ be the point (other than $A$) where the line through $A$ and $P$ intersects the circumcircle of $ABC$. Prove that
\[PQ\le \frac{25}{4\sqrt{6}}.\]
1999 USAMTS Problems, 5
(Revised 2-4-2000) Let $P$ be a point interior to square $ABCD$ so that $PA=a$, $PB=b$, $PC=c$, and $c^2=a^2+2b^2$. Given only the lengths $a$, $b$, and $c$, and using only a compass and a straightedge, construct a square congruent to square $ABCD$.
2004 USAMTS Problems, 4
Region $ABCDEFGHIJ$ consists of $13$ equal squares and is inscribed in rectangle $PQRS$ with $A$ on $\overline{PQ}$, $B$ on $\overline{QR}$, $E$ on $\overline{RS}$, and $H$ on $\overline{SP}$, as shown in the figure on the right. Given that $PQ=28$ and $QR=26$, determine, with proof, the area of region $ABCDEFGHIJ$.
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(12)); pair P=(0,0), Q=(0,28), R=(26,28), S=(26,0), B=(3,28);
draw(P--Q--R--S--cycle);
picture p = new picture;
draw(p, (0,0)--(3,0)^^(0,-1)--(3,-1)^^(0,-2)--(5,-2)^^(0,-3)--(5,-3)^^(2,-4)--(3,-4)^^(2,-5)--(3,-5));
draw(p, (0,0)--(0,-3)^^(1,0)--(1,-3)^^(2,0)--(2,-5)^^(3,0)--(3,-5)^^(4,-2)--(4,-3)^^(5,-2)--(5,-3));
transform t = shift(B) * rotate(-aSin(1/26^.5)) * scale(26^.5);
add(t*p);
label("$P$",P,SW); label("$Q$",Q,NW); label("$R$",R,NE); label("$S$",S,SE); label("$A$",t*(0,-3),W); label("$B$",B,N); label("$C$",t*(3,0),plain.ENE); label("$D$",t*(3,-2),NE); label("$E$",t*(5,-2),plain.E); label("$F$",t*(5,-3),plain.SW); label("$G$",t*(3,-3),(0.81,-1.3)); label("$H$",t*(3,-5),plain.S); label("$I$",t*(2,-5),NW); label("$J$",t*(2,-3),SW);[/asy]
2019 USAMTS Problems, 1
Fill in each empty white circle with a number from $1$ to $16$ so that each number is used exactly once. One number
has been given to you. If a square has a given number inside and its four vertices contain the numbers $a, b, c, d$ in clockwise order, then the number inside the square must be equal to $(a + c)(b + d)$.
There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer 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(1cm);
draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((1,0)--(1,3)); draw((2,0)--(2,3));
string[][] givens = {{"","638","650"},{"50","","338"},{"77","130",""}};
string[][] numbers = {{"","","",""},{"","","",""},{"","","",""},{"","","","5"}};
for(int i=0; i < 4; ++i) {
for(int j=0; j < 4; ++j) {
filldraw(circle((i,j),0.3), white);
label(numbers[3-j][i], (i,j));
}
}
for(int i=0; i < 3; ++i){
for(int j=0; j < 3; ++j){
label(givens[2-j][i], (i + 0.5, j + 0.5));
}
}
[/asy]
2009 USAMTS Problems, 2
Find, with proof, a positive integer $n$ such that
\[\frac{(n + 1)(n + 2) \cdots (n + 500)}{500!}\]
is an integer with no prime factors less than $500$.
1998 USAMTS Problems, 4
Let $A$ consist of $16$ elements of the set $\{1,2,3,\ldots, 106\}$, so that no two elements of $A$ differ by $6, 9, 12, 15, 18,$ or $21$. Prove that two elements of $A$ must differ by $3$.
1999 USAMTS Problems, 4
We will say that an octagon is integral if its is equiangular, its vertices are lattice points (i.e., points with integer coordinates), and its area is an integer. For example, the figure on the right shows an integral octagon of area $21$. Determine, with proof, the smallest positive integer $K$ so that for every positive integer $k\geq K$, there is an integral octagon of area $k$.
[asy]
size(200);
defaultpen(linewidth(0.8));
draw((-1/2,0)--(17/2,0)^^(0,-1/2)--(0,15/2));
for(int i=1;i<=6;++i){
draw((0,i)--(17/2,i),linetype("4 4"));
}
for(int i=1;i<=8;++i){
draw((i,0)--(i,15/2),linetype("4 4"));
}
draw((2,1)--(1,2)--(1,3)--(4,6)--(5,6)--(7,4)--(7,3)--(5,1)--cycle,linewidth(1));
label("$1$",(1,0),S);
label("$2$",(2,0),S);
label("$x$",(17/2,0),SE);
label("$1$",(0,1),W);
label("$2$",(0,2),W);
label("$y$",(0,15/2),NW);
[/asy]
2012 USAMTS Problems, 1
In the $8\times 8$ grid shown, fill in $12$ of the grid cells with the numbers $1-12$ so that the following conditions are satisfied:
[list]
[*]Each cell contains at most one number, and each number from $1-12$ is used exactly once.
[*]Two cells that both contain numbers may not touch, even at a point.
[*]A clue outside the grid pointing at a row or column gives the sum of all the numbers in that row or column. Rows and columns without clues have an unknown sum.[/list]
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]
size(150);
defaultpen(linewidth(0.8));
path arrow=(-1/8,1/8)--(1/8,0)--(-1/8,-1/8)--cycle;
int sumRows[]={3,13,20,0,21,0,18,3};
int sumCols[]={24,1,3,0,20,13,0,11};
for(int i=0;i<=8;i=i+1)
draw((i,0)--(i,8)^^(0,i)--(8,i));
for(int j=0;j<=7;j=j+1)
{
if(sumRows[j]>0)
{
filldraw(shift(-1/4,j+1/2)*arrow,black);
label("$"+(string)sumRows[j]+"$",(-7/8,j+1/2));
}
if(sumCols[j]>0)
{
filldraw(shift(j+1/2,8+3/8)*(rotate(270,origin)*arrow),black);
label("$"+(string)sumCols[j]+"$",(j+1/2,9));
}
}
[/asy]
2024 USAMTS Problems, 3
A sequence of integers $x_1, x_2, \dots, x_k$ is called fibtastic if the difference between any two consecutive elements in the sequence is a Fibonacci number.
The integers from $1$ to $2024$ are split into two groups, each written in increasing order.
Group A is $a_1, a_2, \dots, a_m$ and Group B is $b_1, b_2, \dots, b_n.$
Find the largest integer $M$ such that we can guarantee that we can pick $M$ consecutive elements from either Group A or Group B which form a fibtastic sequence.
As an illustrative example, if a group of numbers is $2, 4, 11, 12, 13, 16, 18, 27, 29, 30,$ the
longest fibtastic sequence is $11, 12, 13, 16, 18,$ which has length $5.$
1999 USAMTS Problems, 1
We define the [i]repetition[/i] number of a positive integer $n$ to be the number of distinct digits of $n$ when written in base $10$. Prove that each positive integer has a multiple which has a repetition number less than or equal to $2$.
2014 USAMTS Problems, 2:
Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.
2012 USAMTS Problems, 4
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. Let $m$ be a positive integer, $m\geq 3$. For every integer $i$ with $1\leq i\leq m$, let \[S_{m,i}=\left\{\left\lfloor\dfrac{2^m-1}{2^{i-1}}n-2^{m-i}+1\right\rfloor\,:\,n=1,2,3,\ldots\right\}.\] For example, for $m=3$,
\begin{align*}S_{3,1}&=\{\lfloor 7n-3\rfloor\,:\,n=1,2,3,\ldots\}
\\&=\{4,11,18,\ldots\},
\\S_{3,2}&=\left\{\left\lfloor\dfrac72n-1\right\rfloor\,:\,n=1,2,3,\ldots\right\}
\\&=\{2,6,9,\ldots\},
\\S_{3,3}&=\left\{\left\lfloor\dfrac74n\right\rfloor\,:\,n=1,2,3,\ldots\right\}
\\&=\{1,3,5,\ldots\}.\end{align*}
Prove that for all $m\geq 3$, each positive integer occurs in exactly one of the sets $S_{m,i}$.
2014 USAMTS Problems, 2:
Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon.
[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705);
filldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8));
draw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle);
pair O = (A_1+A_2+A_3+A_4+A_5)/5;
label("$A_1$",A_1, 2dir(A_1-O));
label("$A_2$",A_2, 2dir(A_2-O));
label("$A_3$",A_3, 2dir(A_3-O));
label("$A_4$",A_4, 2dir(A_4-O));
label("$A_5$",A_5, 2dir(A_5-O));
label("$B_1$",B_1, 2dir(B_1-O));
label("$B_2$",B_2, 2dir(B_2-O));
label("$B_3$",B_3, 2dir(B_3-O));
label("$B_4$",B_4, 2dir(B_4-O));
label("$B_5$",B_5, 2dir(B_5-O));
[/asy]