Found problems: 248
1998 USAMTS Problems, 1
Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$.
2009 USAMTS Problems, 4
Let $a$ and $b$ be positive integers such that all but $2009$ positive integers are expressible in the form $ma + nb$, where $m$ and $n$ are nonnegative integers. If $1776 $is one of the numbers that is not expressible, find $a + b$.
2010 USAMO, 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.
1998 USAMTS Problems, 4
Prove that if $0<x<\pi/2$, then $\sec^6 x+\csc^6 x+(\sec^6 x)(\csc^6 x)\geq 80$.
1999 USAMTS Problems, 5
In a convex pentagon $ABCDE$ the sides have lengths $1,2,3,4,$ and $5$, though not necessarily in that order. Let $F,G,H,$ and $I$ be the midpoints of the sides $AB$, $BC$, $CD$, and $DE$, respectively. Let $X$ be the midpoint of segment $FH$, and $Y$ be the midpoint of segment $GI$. The length of segment $XY$ is an integer. Find all possible values for the length of side $AE$.
1998 USAMTS Problems, 3
The integers from $1$ to $9$ can be arranged into a $3\times3$ array (as shown on the right) so that the sum of the numbers in every row, column, and diagonal is a multiple of $9$.
(a.) Prove that the number in the center of the array must be a multiple of $3$.
(b.) Give an example of such an array with $6$ in the center.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));size(100);
int i,j;
for(i=0; i<4; i=i+1) {
draw((0,2i)--(6,2i));
draw((2i,0)--(2i,6));
}
string[] letters={"G", "H", "I", "D", "E", "F", "A", "B", "C"};
for(i=0; i<3; i=i+1) {
for(j=0; j<3; j=j+1) {
label(letters[3i+j], (2j+1, 2i+1));
}}[/asy]
1999 USAMTS Problems, 5
We say that a finite set of points is [i]well scattered[/i] on the surface of a sphere if every open hemisphere (half the surface of the sphere without its boundary) contains at least one of the points. The set $\{ (1,0,0), (0,1,0), (0,0,1) \}$ is not well scattered on the unit sphere (the sphere of radius $1$ centered at the origin), but if you add the correct point $P$ it becomes well scattered. Find, with proof, all possible points $P$ that would make the set well scattered.
2014 USAMTS Problems, 1:
The net of 20 triangles shown below can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from 1 to 20 with each number used exactly once. Any pair of numbers that are consecutive must be written on faces sharing an edge in the folded icosahedron, and additionally, 1 and 20 must also be on faces sharing an edge. Some numbers have been given to you. (No proof is necessary.)
[asy]
unitsize(1cm);
pair c(int a, int b){return (a-b/2,sqrt(3)*b/2);}
draw(c(0,0)--c(0,1)--c(-1,1)--c(1,3)--c(1,1)--c(2,2)--c(3,2)--c(4,3)--c(4,2)--c(3,1)--c(2,1)--c(2,-1)--c(1,-1)--c(1,-2)--c(0,-3)--c(0,-2)--c(-1,-2)--c(1,0)--cycle);
draw(c(0,0)--c(1,1)--c(0,1)--c(1,2)--c(0,2)--c(0,1),linetype("4 4"));
draw(c(4,2)--c(3,2)--c(3,1),linetype("4 4"));
draw(c(3,2)--c(1,0)--c(1,1)--c(2,1)--c(2,2),linetype("4 4"));
draw(c(1,-2)--c(0,-2)--c(0,-1)--c(1,-1)--c(1,0)--c(2,0)--c(0,-2),linetype("4 4"));
label("2",(c(0,2)+c(1,2))/2,S);
label("15",(c(1,1)+c(2,1))/2,S);
label("6",(c(0,1)+c(1,1))/2,N);
label("14",(c(0,0)+c(1,0))/2,N);[/asy]
2019 USAMTS Problems, 4
Let $FIG$ be a triangle and let $D$ be a point on $\overline{FG}$. The line perpendicular to $\overline{FI}$ passing through the midpoint of $\overline{FD}$ and the line perpendicular to $\overline{IG}$ passing through the midpoint of $\overline{DG}$ intersect at $T$. Prove that $FT = GT$ if and only if $\overline{ID}$ is perpendicular to $\overline{FG}$.
2012 USAMTS Problems, 2
Three wooden equilateral triangles of side length $18$ inches are placed on axles as shown in the diagram to the right. Each axle is $30$ inches from the other two axles. A $144$-inch leather band is wrapped around the wooden triangles, and a dot at the top corner is painted as shown. The three triangles are then rotated at the same speed and the band rotates without slipping or stretching. Compute the length of the path that the dot travels before it returns to its initial position at the top corner.
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(10));
pair A=origin,B=(48,0),C=rotate(60,A)*B;
path equi=(0,0)--(18,0)--(9,9*sqrt(3))--cycle,circ=circle(centroid(A,B,C)*18/48,1/3);
picture a;
fill(a,equi,grey);
fill(a,circ,white);
add(a);
add(shift(15,15*sqrt(3))*a);
add(shift(30,0)*a);
draw(A--B--C--cycle,linewidth(1));
path top = circle(C,2/3);
unfill(top);
draw(top);
real r=-5/2;
draw((9,r+1)--(9,r-1)^^(9,r)--(39,r)^^(39,r-1)--(39,r+1));
label("$30$",(24,r),S);
[/asy]
1998 USAMTS Problems, 5
In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\angle AMN=\angle NMO$. Determine the measure of $\angle A B N$ in degrees.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90);
draw(K--O--A--M--N--B--A^^Circle(origin,1));
label("$A$", A, dir(O--A));
label("$K$", K, dir(O--K));
label("$B$", B, dir(O--B));
label("$N$", N, E);
label("$M$", M, S);
label("$O$", O, SE);[/asy]
2016 Balkan MO Shortlist, N5
A positive integer is called [i]downhill[/i] if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial $P(x)$ with rational coefficients takes on an integer value for each downhill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?
2011 USAMTS Problems, 4
A $\emph{luns}$ with vertices $X$ and $Y$ is a region bounded by two circular arcs meeting at the endpoints $X$ and $Y$. Let $A$, $B$, and $V$ be points such that $\angle AVB=75^\circ$, $AV=\sqrt{2}$ and $BV=\sqrt{3}$. Let $\mathcal{L}$ be the largest area luns with vertices $A$ and $B$ that does not intersect the lines $VA$ or $VB$ in any points other than $A$ and $B$. Define $k$ as the area of $\mathcal{L}$. Find the value \[ \dfrac {k}{(1+\sqrt{3})^2}. \]
2004 USAMTS Problems, 2
Call a number $a-b\sqrt2$ with $a$ and $b$ both positive integers $tiny$ if it is closer to zero than any number $c-d\sqrt2$ such that $c$ and $d$ are positive integers with $c<a$ and $d<b$. Three numbers which are tiny are $1-\sqrt2$, $3-2\sqrt2$, and $7-5\sqrt2$. Without using any calculator or computer, prove whether or not each of the following is tiny:
\[(a)\ 58-41\sqrt2,\qquad\qquad (b)\ 99-70\sqrt2.\]
2009 USAMTS Problems, 5
Tina and Paul are playing a game on a square $S$. First, Tina selects a point $T$ inside $S$. Next, Paul selects a point $P$ inside $S$. Paul then colors blue all the points inside $S$ that are closer to $P$ than $T$ . Tina wins if the blue region thus produced is the interior of a triangle. Assuming that Paul is lazy and simply selects his point at random (and that Tina knows this), find, with proof, a point Tina can select to maximize her probability of winning, and compute this probability.
2004 USAMTS Problems, 1
Given two integers $x$ and $y$, let $(x \| y)$ denote the [i]concatenation[/i] of $x$ by $y$, which is obtained by appending the digits of $y$ onto the end of $x$. For example, if $x=218$ and $y=392$, then $(x \| y) = 218392$.
(a) Find 3-digit integers $x$ and $y$ such that $6(x \| y) = (y \| x)$.
(b) Find 9-digit integers $x$ and $y$ such that $6(x \| y) = (y \| x)$.
2013 USAMTS Problems, 2
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.
2009 USAMTS Problems, 1
Jeremy has a magic scale, each side of which holds a positive integer. He plays the following game: each turn, he chooses a positive integer $n$. He then adds $n$ to the number on the left side of the scale, and multiplies by $n$ the number on the right side of the scale. (For example, if the turn starts with $4$ on the left and $6$ on the right, and Jeremy chooses $n = 3$, then the turn ends with $7$ on the left and $18$ on the right.) Jeremy wins if he can make both sides of the scale equal.
(a) Show that if the game starts with the left scale holding $17$ and the right scale holding $5$, then Jeremy can win the game in $4$ or fewer turns.
(b) Prove that if the game starts with the right scale holding $b$, where $b\geq 2$, then Jeremy can win the game in $b-1$ or fewer turns.
1998 USAMTS Problems, 3
Let $f$ be a polynomial of degree $98$, such that $f (k) =\frac{1}{k}$ for $k=1,2,3,\ldots,99$. Determine $f(100)$.
2017 USAMTS Problems, 5
There are $n$ distinct points in the plane, no three of which are collinear. Suppose that $A$ and $B$ are two of these points. We say that segment $AB$ is independent if there is a straight line such that points $A$ and $B$ are on one side of the line, and the other $n-2$ points are on the other side. What is the maximum possible number of independent segments?
2012 USAMTS Problems, 2
Palmer and James work at a dice factory, placing dots on dice. Palmer builds his dice correctly, placing the dots so that $1$, $2$, $3$, $4$, $5$, and $6$ dots are on separate faces. In a fit of mischief, James places his $21$ dots on a die in a peculiar order, putting some nonnegative integer number of dots on each face, but not necessarily in the correct con figuration. Regardless of the confi guration of dots, both dice are unweighted and have equal probability of showing each face after being rolled.
Then Palmer and James play a game. Palmer rolls one of his normal dice and James rolls his peculiar die. If they tie, they roll again. Otherwise the person with the larger roll is the winner. What is the maximum probability that James wins? Give one example of a peculiar die that attains this maximum probability.
1999 USAMTS Problems, 4
We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths.
(a) Find an integral triangle with perimeter of $42$.
(b) Is there an integral triangle with perimeter of $43$?
2020 USAMTS Problems, 3:
Given a nonconstant polynomial with real coefficients $f(x),$ let $S(f)$ denote the sum of its roots. Let p and q be nonconstant polynomials with real coefficients such that $S(p) = 7,$ $S(q) = 9,$ and $S(p-q)= 11$. Find, with proof, all possible values for $S(p + q)$.
2002 USAMTS Problems, 1
The integer $n$, between 10000 and 99999, is $abcde$ when written in decimal notation. The digit $a$ is the remainder when $n$ is divided by 2, the digit $b$ is the remainder when $n$ is divided by 3, the digit $c$ is the remainder when $n$ is divided by 4, the digit $d$ is the remainder when $n$ is divied by 5, and the digit $e$ is the reminader when $n$ is divided by 6. Find $n$.
2011 USAMTS Problems, 3
Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.