Found problems: 248
2013 USAMTS Problems, 5
Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much?
2018 USAMTS Problems, 3:
Find, with proof, all pairs of positive integers $(n,d)$ with the following property: for every integer $S$, there exists a unique non-decreasing sequence of $n$ integers $a_1,a_2,...,a_n$ such that $a_1 + a_2 + ... + a_n = S$ and $a_n-a_1=d.$
2013 USAMTS Problems, 4
Bunbury the bunny is hopping on the positive integers. First, he is told a positive integer $n$. Then Bunbury chooses positive integers $a,d$ and hops on all of the spaces $a,a+d,a+2d,\dots,a+2013d$. However, Bunbury must make these choices so that the number of every space that he hops on is less than $n$ and relatively prime to $n$.
A positive integer $n$ is called [i]bunny-unfriendly[/i] if, when given that $n$, Bunbury is unable to find positive integers $a,d$ that allow him to perform the hops he wants. Find the maximum bunny-unfriendly integer, or prove that no such maximum exists.
2016 USAMTS Problems, 2:
A tower of height $h$ is a stack of contiguous rows of squares of height $h$ such that
[list]
[*] the bottom row of the tower has $h$ squares,
[*] each row above the bottom row has one fewer square than the row below it, and within each row the squares are contiguous,
[*] the squares in any given row all lie directly above a square in the row below.
[/list]
A tower is called balanced if when the squares of the tower are colored black and white in a checkerboard fashion, the number of black squares is equal to the number of white squares. For example, the figure above shows a tower of height 5 that is not balanced, since there are 7 white squares and 8 black squares.
How many balanced towers are there of height 2016?
2011 AMC 10, 24
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
$ \textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{\sqrt{2}}{12}\qquad\textbf{(C)}\ \frac{\sqrt{3}}{12}\qquad\textbf{(D)}\ \frac{1}{6}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{6} $
2008 AIME Problems, 8
Find the positive integer $ n$ such that \[\arctan\frac{1}{3}\plus{}\arctan\frac{1}{4}\plus{}\arctan\frac{1}{5}\plus{}\arctan\frac{1}{n}\equal{}\frac{\pi}{4}.\]
2014 USAMTS Problems, 4:
Let $\omega_P$ and $\omega_Q$ be two circles of radius $1$, intersecting in points $A$ and $B$. Let $P$ and $Q$ be two regular $n$-gons (for some positive integer $n\ge4$) inscribed in $\omega_P$ and $\omega_Q$, respectively, such that $A$ and $B$ are vertices of both $P$ and $Q$. Suppose a third circle $\omega$ of radius $1$ intersects $P$ at two of its vertices $C$, $D$ and intersects $Q$ at two of its vertices $E$, $F$. Further assume that $A$, $B$, $C$, $D$, $E$, $F$ are all distinct points, that $A$ lies outside of $\omega$, and that $B$ lies inside $\omega$. Show that there exists a regular $2n$-gon that contains $C$, $D$, $E$, $F$ as four of its vertices.
2017 USAMTS Problems, 2
Let $b$ be a positive integer. Grogg writes down a sequence whose first term is $1$. Each term after that is the total number of digits in all the previous terms of the sequence when written in base $b$. For example, if $b = 3$, the sequence starts $1, 1, 2, 3, 5, 7, 9, 12, \dots$. If $b = 2521$, what is the first positive power of $b$ that does not appear in the sequence?
2004 USAMTS Problems, 5
Two circles of equal radius can tightly fit inside right triangle $ABC$, which has $AB=13$, $BC=12$, and $CA=5$, in the three positions illustrated below. Determine the radii of the circles in each case.
[asy]
size(400); defaultpen(linewidth(0.7)+fontsize(12)); picture p = new picture; pair s1 = (20,0), s2 = (40,0); real r1 = 1.5, r2 = 10/9, r3 = 26/7; pair A=(12,5), B=(0,0), C=(12,0);
draw(p,A--B--C--cycle); label(p,"$B$",B,SW); label(p,"$A$",A,NE); label(p,"$C$",C,SE);
add(p); add(shift(s1)*p); add(shift(s2)*p);
draw(circle(C+(-r1,r1),r1)); draw(circle(C+(-3*r1,r1),r1));
draw(circle(s1+C+(-r2,r2),r2)); draw(circle(s1+C+(-r2,3*r2),r2));
pair D=s2+156/17*(A-B)/abs(A-B), E=s2+(169/17,0), F=extension(D,E,s2+A,s2+C);
draw(incircle(s2+B,D,E)); draw(incircle(s2+A,D,F));
label("Case (i)",(6,-3)); label("Case (ii)",s1+(6,-3)); label("Case (iii)",s2+(6,-3));[/asy]
1999 USAMTS Problems, 4
There are $8436$ steel balls, each with radius $1$ centimeter, stacked in a tetrahedral pile, with one ball on top, $3$ balls in the second layer, $6$ in the third layer, $10$ in the fourth, and so on. Determine the height of the pile in centimeters.
2002 USAMTS Problems, 5
A fudgeflake is a planar fractal figure with a $120^{\circ}$ rotational symmetry such that three identical fudgeflakes in the same orientation fit together without gaps to form a larger fudgeflake with its orientation $30^{\circ}$ clockwise of the smaller fudgeflakes' orientation, as shown below. If the distance between the centers of the original three fudgeflakes is $1$, what is the area of one of those three fudgeflakes? Justify your answer.
[asy]
defaultpen(linewidth(.7));
string s = "00d08c8520612022202288272220065886,00e0708768822888788866683,01006c8765,01206b88227606,01208c858768588616678868160027,017068870228728868822872272220600278886,0190988886872272882228166060,02209486868506,02304b282022852282022828888828228166060066002000668666,02304d8882858688886166702,023050,0230637222282272220786,0240918681,02505f,0260908527222027285886852728522820766,02905b81,03904422888888766686882288888607,03908c58588685876668688228882078688228822220258587685886166702,049053852282000027888666,0490fa852061112222282282000702202220065868,04a0ae868822888888866602768688228888860728228822820228586888766702,04a0de2821666868822888500602,04b0ac5812022882227200070,04c04a8220228822272012882876882288227606,04c0da288282220228886882288227600660020060668,04c0f2,04e0fa88868588616668688228882078688228822220786,05e0ba878605,05e0d287688872006,05e102786720,05f0b88220228822816606066860,06002786882288886888666868886166600668666868887,0600748207,0600ce88616,0600ff86816,0610258222282220228887882288228202285868886668688228858688228822827066,062023223,0620cd88227600,0620fe8527222027686,064074555555555555555555555555555555555552882288888876668688866606060276866686882288878886668668522888868527282822816660222228588688861668678886166600668666868886660602782228,0650c985877,0680b0865282888882822202288878886668678282858768522882822200602788860660606,06a0fa88886070220,06c0b48830,0700fe8527222206702,0790fb867888666868822888788228722722720128288768886668688866600668666027686882288886888606786882288888607,0870b9822202288222022888886058,0880428582288868886668688228828768822882282166606060602,088070816,08903e7870,08a06c877,08b02a82858868886067222006,08b03b867202285272220786,08c06b8528220676,09003572,0900745555555555555555555555555555555555555555555555555555555555555555555555555558207,0920668858888285272220786,09606c,09812e88228166,09a128868827,09b12682202288222766060061,09c0648867,09c094868672020228160,09d04b86787228828868886668688866616006686605820228,09d0e38207,09e0908282822202288888828282220228886888681666868522888888285228202282886685886066886888606606666678868160027,09f0478765,0a10468822220276866606,0a411e828886882288227606060602,0a90428888681288227600,0aa046,0aa091,0b111f8282886788766866852288586882288228216660606,0b2122,0b30438861,0b5042858868886668688228882078688228822760600660,0b805f28868686868686850633,0be09d,0c3127868681020222022882276066660,0c40de86788228728868886668688866606066860678688228886888681666868527285886872272882228166602,0c504b88688810600220222027686,0c60da86816,0c70a8,0c80d98527222027686,0c90b08886888666868822888886058228822820228287688868166768886816606006786667021,0cb046827222025888685272850,0cc11f8702288888605822882281660602,0d00d588886070220,0d60d98527222070070,0d811a86868850607,0d90b8786783,0da0268768822882876888666868886661600668666868886816611,0da107272858868886166786888616660066870202,0db0b586816,0dc02527201288227600,0dd0b48527222027686,0e301e8861,0e501d88888605822882222027686,0e50b0888860702,0eb0b48527222206702,0f401d88868886166702,0f40b1867888666868822888788228822202786,0f50298822882888860652288227600,0fc02d,1010b98765,10206f82222282220228888882828222006588606600660020066660660066868,10207288228888685866686888681636160706768166686882288878886706786882288868822885228166,10303e28216668688228887685886166702,1030b8220258527227,10502b87,10503a288282220065886066006,10602958882078688866683,10904e8527285886882288227600,10a0b127282887685272858122816,1160ad86885,11706327683,1170ab8216,11904f877,11905f87022872812220605886,119097882886888666868886661600658,11b04e858868886166783";
string[] k = split(s,",");
for(string str:k) {
string a = substr(str,0,3),b=substr(str,3,3),c=substr(str,6);
real x=hex(a),y=hex(b);
for (int i=0;i<length(c);++i) {
int next = hex(substr(c,i,1));
real yI=(int)((next-next%3)/3),xI=next%3;
--xI; --yI;
draw((x,-y)--(x+xI,-(y+yI)));
x+=xI;
y+=yI;
}
}[/asy]
1999 USAMTS Problems, 3
The figure on the right shows the map of Squareville, where each city block is of the same length. Two friends, Alexandra and Brianna, live at the corners marked by $A$ and $B$, respectively. They start walking toward each other's house, leaving at the same time, walking with the same speed, and independently choosing a path to the other's house with uniform distribution out of all possible minimum-distance paths [that is, all minimum-distance paths are equally likely]. What is the probability they will meet?
[asy]
size(200);
defaultpen(linewidth(0.8));
for(int i=0;i<=2;++i) {
for(int j=0;j<=4;++j) {
draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle);
}
}
for(int i=3;i<=4;++i) {
for(int j=3;j<=6;++j) {
draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle);
}
}
label("$A$",origin,SW);
label("$B$",(5,7),SE);
[/asy]
2017 USAMTS Problems, 3
Do there exist two polygons such that, by putting them together in three different ways (without holes, overlap, or reflections), we can obtain first a triangle, then a convex quadrilateral, and lastly a convex pentagon?
2002 USAMTS Problems, 4
The vertices of a cube have coordinates $(0,0,0),(0,0,4),(0,4,0),(0,4,4),(4,0,0)$,$(4,0,4),(4,4,0)$, and $(4,4,4)$. A plane cuts the edges of this cube at the points $(0,2,0),(1,0,0),(1,4,4)$, and two other points. Find the coordinates of the other two points.
2009 USAMTS Problems, 2
Let $a, b, c, d$ be four real numbers such that
\begin{align*}a + b + c + d &= 8, \\
ab + ac + ad + bc + bd + cd &= 12.\end{align*}
Find the greatest possible value of $d$.
2015 USAMTS Problems, 5
Let $n>1$ be an even positive integer. An $2n \times 2n$ grid of unit squares is given, and it is partitioned into $n^2$ contiguous $2 \times 2$ blocks of unit squares. A subset $S$ of the unit squares satisfies the following properties:
(i) For any pair of squares $A,B$ in $S$, there is a sequence of squares in $S$ that starts with $A$, ends with $B$, and has any two consecutive elements sharing a side; and
(ii) In each of the $2 \times 2$ blocks of squares, at least one of the four squares is in $S$.
An example for $n=2$ is shown below, with the squares of $S$ shaded and the four $2 \times 2$ blocks of squares outlined in bold.
[asy]
size(2.5cm);
fill((0,0)--(4,0)--(4,1)--(0,1)--cycle,mediumgrey);
fill((0,0)--(0,4)--(1,4)--(1,0)--cycle,mediumgrey);
fill((0,3)--(4,3)--(4,4)--(0,4)--cycle,mediumgrey);
fill((3,0)--(3,4)--(4,4)--(4,0)--cycle,mediumgrey);
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((1,0)--(1,4));
draw((2,0)--(2,4),linewidth(1));
draw((3,0)--(3,4));
draw((0,1)--(4,1));
draw((0,2)--(4,2),linewidth(1));
draw((0,3)--(4,3));
[/asy]
In terms of $n$, what is the minimum possible number of elements in $S$?
2005 USAMTS Problems, 4
Find, with proof, all irrational numbers $x$ such that both $x^3-6x$ and $x^4-8x^2$ are rational.
2020 USAMTS Problems, 1:
Place the 21 two-digit prime numbers in the white squares of the grid on the right so that each two-digit prime is used exactly once. Two white squares sharing a side must contain two numbers with either the same tens digit or ones digit. A given digit in a white square must equal at least one of the two digits of that square’s prime number.
[asy]
size(10cm);
real s= 10.0;
int[][] x = {
{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}};
void square(int a, int b)
{
fill(s*(a,b)--s*(a+1,b)--s*(a+1,b+1)--s*(a,b+1)--cycle);
}
square(1,2);
square(1,3);
square(3,1);
square(3,2);
for(int i = 0; i < 6; ++i) {
draw(s*(i,0)--s*(i,5));
}
for(int i = 0; i < 6; ++i) {
draw(s*(0,i)--s*(5,i));
}
for(int k = 0; k<5; ++k){
for(int l = 0; l<5; ++l){
if(x[k][l]!=0){
label(scale(5.0)*string(x[k][l]),s*(l+0.5,-k+4.5));
}
}
}
void sudokuLabel(int p, int q, int r) {
label(string(r), s*(p, q) + (1, -1));
}
sudokuLabel(1, 1, 4);
sudokuLabel(2, 1, 1);
sudokuLabel(3, 1, 1);
sudokuLabel(4, 1, 3);
sudokuLabel(0, 3, 9);
sudokuLabel(2, 3, 9);
sudokuLabel(4, 3, 5);
sudokuLabel(0, 5, 3);
sudokuLabel(1, 5, 1);
sudokuLabel(2, 5, 3);
sudokuLabel(3, 5, 2);[/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.)
2017 USAMTS Problems, 5
Let $n$ be a positive integer. Aavid has a card deck consisting of $ 2n$ cards, each colored with one of $n$ colors such that every color is on exactly two of the cards. The $2n$ cards are randomly ordered in a stack. Every second, he removes the top card from the stack and places the card into an area called the pit. If the other card of that color also happens to be in the pit, Aavid collects both cards of that color and discards them from the pit.
Of the $(2n)!$ possible original orderings of the deck, determine how many have the following property: at every point, the pit contains cards of at most two distinct colors.
2021 USAMTS Problems, 2
Sydney the squirrel is at $(0, 0)$ and is trying to get to $(2021, 2022)$. She can move only by reflecting her position over any line that can be formed by connecting two lattice points, provided that the reflection puts her on another lattice point. Is it possible for Sydney to reach $(2021, 2022)$?
2017 USAMTS Problems, 2
Let $q$ be a real number. Suppose there are three distinct positive integers $a, b,c$ such that $q + a$, $q + b$,$q + c$ is a geometric progression. Show that $q$ is rational.
2014 USAMTS Problems, 2:
Let $a, b, c, x$ and $y$ be positive real numbers such that $ax + by \leq bx + cy \leq cx + ay$.
Prove that $b \leq c$.
1998 USAMTS Problems, 2
For a nonzero integer $i$, the exponent of $2$ in the prime factorization of $i$ is called $ord_2 (i)$. For example, $ord_2(9)=0$ since $9$ is odd, and $ord_2(28)=2$ since $28=2^2\times7$. The numbers $3^n-1$ for $n=1,2,3,\ldots$ are all even so $ord_2(3^n-1)>0$ for $n>0$.
a) For which positive integers $n$ is $ord_2(3^n-1) = 1$?
b) For which positive integers $n$ is $ord_2(3^n-1) = 2$?
c) For which positive integers $n$ is $ord_2(3^n-1) = 3$?
Prove your answers.