Found problems: 248
2020 USAMTS Problems, 1:
In the grid below, fill each gray cell with one of the numbers from the provided bank, with each number used once, and fill each white cell with a positive one-digit number. The number in a gray cell must equal the sum of the numbers in all touching white cells, where two cells sharing a vertex are considered touching. All of the terms in each of these sums must be distinct, meaning that two white cells with the same digit may not touch the same gray cell.
Bank: 15, 23, 28, 35, 36, 38, 40, 42, 44
[asy]
unitsize(1.2cm);
defaultpen(fontsize(30pt));
int[][] x = {
{0,5,0,0,0,8,0},
{4,0,0,0,0,0,0},
{0,3,0,0,0,0,1},
{0,0,0,0,0,7,0},
{0,8,0,0,1,0,4},
{0,0,0,0,0,2,0}};
void square(int a, int b)
{
filldraw((a,b)--(a+1,b)--(a+1,b+1)--(a,b+1)--cycle,mediumgray);
}
square(1,0);
square(3,1);
square(5,1);
square(1,2);
square(3,3);
square(5,3);
square(1,4);
square(5,4);
square(3,5);
for(int i = 0; i < 8; ++i) {
draw((i,0)--(i,6));
}
for(int i = 0; i < 7; ++i) {
draw((0,i)--(7,i));
}
for(int k = 0; k<6; ++k){
for(int l = 0; l<7; ++l){
if(x[k][l]!=0){
label(string(x[k][l]),(l+0.5,-k+5.5));
}
}
}
[/asy]
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.)
2015 USAMTS Problems, 1
In the grid to the right, the shortest path through unit squares between between the pair of 2's has length 2. Fill in some of the unit squares in the grid so that
(i) exactly half of the squares in each row and column contain a number,
(ii) each of the number 1 through 12 appears exactly twice, and
(iii) for $n=1,2,\cdot\cdot\cdot,12$, the shortest path between the pair of $n$'s has length exactly $n$.
1999 USAMTS Problems, 5
In $\triangle ABC$, $AC>BC$, $CM$ is the median, and $CH$ is the altitude emanating from $C$, as shown in the figure on the right. Determine the measure of $\angle MCH$ if $\angle ACM$ and $\angle BCH$ each have measure $17^\circ$.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=origin,B=(10,0),C=(7,5),M=(5,0),H=(7,0);
draw(A--C--B--cycle^^H--C--M);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,NE);
label("$M$",M,NW);
label("$H$",H,NE);
[/asy]
2004 USAMTS Problems, 2
Find positive integers $a$, $b$, and $c$ such that
\[\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{219+\sqrt{10080}+\sqrt{12600}+\sqrt{35280}}.\]
Prove that your solution is correct. (Warning: numerical approximations of the values do not constitute a proof.)
2021 USAMTS Problems, 5
For a finite nonempty set $A$ of positive integers, $A =\{a_1, a_2,\dots , a_n\}$, we say the calamitous complement of A is the set of all positive integers $k$ for which there do not exist nonnegative integers $w_1, w_2, \dots , w_n$ with $k = a_1w_1 + a_2w_2 +\dots + a_nw_n.$ The calamitous complement of $A$ is denoted $cc(A)$. For example, $cc(\{5, 6, 9\}) = \{1, 2, 3, 4, 7, 8, 13\}$.
Find all pairs of positive integers $a, b$ with $1 < a < b$ for which there exists a set $G$ satisfying all of the following properties:
1. $G$ is a set of at most three positive integers,
2. $cc(\{a, b\})$ and $cc(G)$ are both finite sets, and
3. $cc(G) = cc(\{a, b\})\cup \{m\}$ for some $m$ not in $cc(\{a, b\})$.
2016 USAMTS Problems, 5:
Let $ABCD$ be a convex quadrilateral with perimeter $\tfrac{5}{2}$ and $AC=BD=1$. Determine the maximum possible area of $ABCD$.
2017 USAMTS Problems, 4
A positive integer is called [i]uphill [/i] if the digits in its decimal representation form an increasing sequence from left to right. That is, a number $\overline{a_1a_2... a_n}$ is uphill if $a_i \le a_{i+1}$ for all $i$. For example, $123$ and $114$ are both uphill. Suppose a polynomial $P(x)$ with rational coefficients takes on an integer value for each uphill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?
2012 USAMTS Problems, 3
In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.
2017 USAMTS Problems, 1
Fill in each cell of the grid with a positive digit so that the following conditions hold:
1. each row and column contains ve distinct digits;
2. for any cage containing multiple cells of a row, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole row, and
3. for any cage containing multiple cells of a column, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole column.
You do not need to prove that your answer is the only one possible; you merely need to fi nd an answer that satis fies 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(48);
int[][] a = {
{0, 0, 0, 0, 0},
{0, 0, 0, 0, 0},
{0, 0, 0, 0, 0},
{0, 0, 0, 0, 0},
{0, 0, 0, 0, 0}};
for (int i = 0; i < 5; ++i) {
for (int j = 0; j < 5; ++j) {
draw((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(24pt));
}
}
real ep=0.1;
real s=3;
pen lw=linewidth(.12mm);
real x=0.9;
real y=1.2;
draw((0+s*ep,0-ep)--(2-ep,0-ep)--(2-ep,-1+ep)--(0+ep,-1+ep)--(0+ep,0-s*ep),dashed+lw);
draw((2+s*ep,0-ep)--(4-ep,0-ep)--(4-ep,-1+ep)--(2+ep,-1+ep)--(2+ep,0-s*ep),dashed+lw);
draw((1+s*ep,-1-ep)--(3-ep,-1-ep)--(3-ep,-2+ep)--(1+ep,-2+ep)--(1+ep,-1-s*ep),dashed+lw);
draw((2+s*ep,-3-ep)--(4-ep,-3-ep)--(4-ep,-4+ep)--(2+ep,-4+ep)--(2+ep,-3-s*ep),dashed+lw);
draw((1+s*ep,-4-ep)--(3-ep,-4-ep)--(3-ep,-5+ep)--(1+ep,-5+ep)--(1+ep,-4-s*ep),dashed+lw);
draw((3+s*ep,-4-ep)--(5-ep,-4-ep)--(5-ep,-5+ep)--(3+ep,-5+ep)--(3+ep,-4-s*ep),dashed+lw);
label(scale(x)*"5", (0+ep,0-y*ep));
label(scale(x)*"7", (2+ep,0-y*ep));
label(scale(x)*"10", (1+ep,-1-y*ep));
label(scale(x)*"5", (2+ep,-3-y*ep));
label(scale(x)*"2", (1+ep, -4-y*ep));
label(scale(x)*"13", (3+ep, -4-y*ep));
draw((4+s*ep,0-ep)--(5-ep,0-ep)--(5-ep,-2+ep)--(4+ep,-2+ep)--(4+ep,0-s*ep),dashed+lw);
draw((0+s*ep,-1-ep)--(1-ep,-1-ep)--(1-ep,-3+ep)--(0+ep,-3+ep)--(0+ep,-1-s*ep),dashed+lw);
draw((3+s*ep,-1-ep)--(4-ep,-1-ep)--(4-ep,-3+ep)--(3+ep,-3+ep)--(3+ep,-1-s*ep),dashed+lw);
draw((0+s*ep,-3-ep)--(1-ep,-3-ep)--(1-ep,-5+ep)--(0+ep,-5+ep)--(0+ep,-3-s*ep),dashed+lw);
draw((1+s*ep,-2-ep)--(2-ep,-2-ep)--(2-ep,-4+ep)--(1+ep,-4+ep)--(1+ep,-2-s*ep),dashed+lw);
draw((4+s*ep,-2-ep)--(5-ep,-2-ep)--(5-ep,-4+ep)--(4+ep,-4+ep)--(4+ep,-2-s*ep),dashed+lw);
label(scale(x)*"10", (4+ep,0-y*ep));
label(scale(x)*"3", (0+ep,-1-y*ep));
label(scale(x)*"8", (3+ep,-1-y*ep));
label(scale(x)*"16", (1+ep,-2-y*ep));
label(scale(x)*"6", (4+ep,-2-y*ep));
label(scale(x)*"11", (0+ep,-3-y*ep));
[/asy]
2015 USAMTS Problems, 5
Let $a_1,a_2,\dots,a_{100}$ be a sequence of integers. Initially, $a_1=1$, $a_2=-1$ and the remaining numbers are $0$. After every second, we perform the following process on the sequence: for $i=1,2,\dots,99$, replace $a_i$ with $a_i+a_{i+1}$, and replace $a_{100}$ with $a_{100}+a_1$. (All of this is done simultaneously, so each new term is the sum of two terms of the sequence from before any replacements.) Show that for any integer $M$, there is some index $i$ and some time $t$ for which $|a_i|>M$ at time $t$.
2014 USAMTS Problems, 3b:
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in [i]almost-order[/i] with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(b) How many different ways are there to line up $20$ people in [i]almost-order[/i] if their heights are $120, 125, 130,$ $135,$ $140,$ $145,$ $150,$ $155,$ $160,$ $164, 165, 170, 175, 180, 185, 190, 195, 200, 205$, and $210$ centimeters? (Note that there is someone of height $164$ centimeters.)
1998 USAMTS Problems, 2
Determine the smallest rational number $\frac{r}{s}$ such that $\frac{1}{k}+\frac{1}{m}+\frac{1}{n}\leq \frac{r}{s}$ whenever $k, m,$ and $n$ are positive integers that satisfy the inequality $\frac{1}{k}+\frac{1}{m}+\frac{1}{n} < 1$.
2004 USAMTS Problems, 3
A set is $reciprocally\ whole$ if its elements are distinct integers greater than 1 and the sum of the reciprocals of all these elements is exactly 1. Find a set $S$, as small as possible, that contains two reciprocally whole subsets, $I$ and $J$, which are distinct, but not necessarily disjoint (meaning they may share elements, but they may not be the same subset). Prove that no set with fewer elements than $S$ can contain two reciprocally whole subsets.
2012 Putnam, 2
Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$
2020 USAMTS Problems, 2:
Find distinct points $A, B, C,$ and $D$ in the plane such that the length of the segment $AB$ is an even integer, and the lengths of the segments $AC, AD, BC, BD,$ and $CD$ are all odd integers. In addition to stating the coordinates of the points and distances between points, please include a brief explanation of how you found the configuration of points and computed the distances.
2009 USAMTS Problems, 3
I give you a deck of $n$ cards numbered $1$ through $n$. On each turn, you take the top card of the deck and place it anywhere you choose in the deck. You must arrange the cards in numerical order, with card $1$ on top and card $n$ on the bottom. If I place the deck in a random order before giving it to you, and you know the initial order of the cards, what is the expected value of the minimum number of turns you need to arrange the deck in order?
2016 USAMTS Problems, 3:
Find all positive integers $n$ for which $(x^n+y^n+z^n)/2$ is a perfect square whenever $x$, $y$, and $z$ are integers such that $x+y+z=0$.
2016 USAMTS Problems, 2:
Find all triples of three-digit positive integers $x < y < z$ with $x,y,z$ in arithmetic progression and $x, y, z + 1000$ in geometric progression.
[i]For this problem, you may use calculators or computers to gain an intuition about how to solve the problem. However, your final submission should include mathematical derivations or proofs and should not be a solution by exhaustive search.[/i]
2014 USAMTS Problems, 5:
A finite set $S$ of unit squares is chosen out of a large grid of unit squares. The squares of $S$ are tiled with isosceles right triangles of hypotenuse $2$ so that the triangles do not overlap each other, do not extend past $S$, and all of $S$ is fully covered by the triangles. Additionally, the hypotenuse of each triangle lies along a grid line, and the vertices of the triangles lie at the corners of the squares. Show that the number of triangles must be a multiple of $4$.
2012 USAMTS Problems, 1
Several children were playing in the ugly tree when suddenly they all fell.
$\bullet$ Roger hit branches $A$, $B$, and $C$ in that order on the way down.
$\bullet$ Sue hit branches $D$, $E$, and $F$ in that order on the way down.
$\bullet$ Gillian hit branches $G$, $A$, and $C$ in that order on the way down.
$\bullet$ Marcellus hit branches $B$, $D$, and $H$ in that order on the way down.
$\bullet$ Juan-Phillipe hit branches $I$, $C$, and $E$ in that order on the way down.
Poor Mikey hit every branch A through $I$ on the way down. Given only this information, in how many different orders could he have hit these 9 branches on the way down?
2015 USAMTS Problems, 3
For $n > 1$, let $a_n$ be the number of zeroes that $n!$ ends with when written in base $n$. Find the maximum value of $\frac{a_n}{n}$.
2008 AIME Problems, 6
The sequence $ \{a_n\}$ is defined by
\[ a_0 \equal{} 1,a_1 \equal{} 1, \text{ and } a_n \equal{} a_{n \minus{} 1} \plus{} \frac {a_{n \minus{} 1}^2}{a_{n \minus{} 2}}\text{ for }n\ge2.
\]The sequence $ \{b_n\}$ is defined by
\[ b_0 \equal{} 1,b_1 \equal{} 3, \text{ and } b_n \equal{} b_{n \minus{} 1} \plus{} \frac {b_{n \minus{} 1}^2}{b_{n \minus{} 2}}\text{ for }n\ge2.
\]Find $ \frac {b_{32}}{a_{32}}$.
2024 USAMTS Problems, 1
The ``Manhattan distance" between two cells is the shortest distance between those cells when traveling up, down, left, or right, as if one were traveling along city blocks rather than as the crow flies.
Place numbers from $1$-$6$ in some cells so the following criteria are satisfied:
$1.$ A cell contains at most one number. Cells can be left empty.
$2.$ For each cell containing a number $N$ in the grid, exactly two other cells containing $N$ are at a Manhattan distance of $N.$
$3.$ For each cell containing a number $N$ in the grid, no other cells containing $N$ are at a Manhattan distance less than $N.$
[asy]
//credits to fruitmonster97 for the diagram
unitsize(1.25cm);
//The gridlines
for(int i=-3;i<4;++i){
draw((i,3)--(i,-3),lightgray+linewidth(1));
}
for(int j=-3;j<4;++j){
if (j==0){
draw((4,j)--(-4,j),lightgray+linewidth(1));
}else{
draw((3,j)--(-3,j),lightgray+linewidth(1));
}
}
//The outline
draw((-4,-1)--(-4,1)--(-3,1)--(-3,3)--(3,3)--(3,1)--(4,1)--(4,-1)--(3,-1)--(3,-3)--(-3,-3)--(-3,-1)--cycle);
//The numbers
label("$1$",(-0.5,0.5));
label("$1$",(0.5,-0.5));
label("$2$",(-1.5,1.5));
label("$2$",(-2.5,-1.5));
label("$3$",(2.5,1.5));
label("$3$",(-3.5,0.5));
label("$4$",(3.5,-0.5));
label("$4$",(1.5,-2.5));
label("$4$",(-1.5,2.5));
label("$5$",(-2.5,1.5));
label("$5$",(2.5,-1.5));
label("$6$",(1.5,-1.5));
[/asy]
2023 USAMTS Problems, 3
Let $n \geq 2$ be a positive integer, and suppose buildings of height $1, 2, \ldots, n$ are built
in a row on a street. Two distinct buildings are said to be $\emph{roof-friendly}$ if every building
between the two is shorter than both buildings in the pair. For example, if the buildings are
arranged $5, 3, 6, 2, 1, 4,$ there are $8$ roof-friendly pairs: $(5, 3), (5, 6), (3, 6), (6, 2), (6, 4), (2, 1),$
$(2, 4), (1, 4).$ Find, with proof, the minimum and maximum possible number of roof-friendly
pairs of buildings, in terms of $n.$
2017 USAMTS Problems, 1
Given a rectangular grid with some cells containing one letter, we say a row or column is [i]edible [/i] if it has more than one cell with a letter and all such cells contain the same letter. Given such a grid, the hungry, hungry letter monster repeats the following procedure: he nds all edible rows and all edible columns and simultaneously eats all the letters in those rows and columns, removing those letters from the grid and leaving those cells empty. He continues this until no more edible rows and columns remain. Call a grid a [i]meal [/i] if the letter monster can eat all of its letters using this procedure.
In the $7$ by $7$ grid to the right, ll each empty space with one letter so that the grid is a meal and there are a total of eight Us, nine Ss, ten As, eleven Ms, and eleven Ts. Some letters have been given to you.
You do not need to prove that your answer is the only one possible; you merely need to find an answer that satis fies 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.)
[img]https://cdn.artofproblemsolving.com/attachments/9/a/d1886720796e4befd9d3ce0cbd2868d1b649d1.png[/img]