Found problems: 26
2022 BAMO, E/3
A polygon is called [i]convex[/i] if all its internal angles are smaller than 180$^{\circ}$. Given a convex polygon, prove that one can find three distinct vertices $A$, $P$, and $Q$, where $PQ$ is a side of the polygon, such that the perpendicular from $A$ to the line $PQ$ meets the segment $PQ$ (possible at $P$ of $Q$).
2017 BAMO, C/1
Find all natural numbers $n$ such that when we multiply all divisors of $n$, we will obtain $10^9$. Prove that your number(s) $n$ works and that there are no other such numbers.
([i]Note[/i]: A natural number $n$ is a positive integer; i.e., $n$ is among the counting numbers $1, 2, 3, \dots$. A [i]divisor[/i] of $n$ is a natural number that divides $n$ without any remainder. For example, $5$ is a divisor of $30$ because $30 \div 5 = 6$; but $5$ is not a divisor of $47$ because $47 \div 5 = 9$ with remainder $2$. In this problem, we consider only positive integer numbers $n$ and positive integer divisors of $n$. Thus, for example, if we multiply all divisors of $6$ we will obtain $36$.)
1999 BAMO, 2
Let $O = (0,0), A = (0,a), and B = (0,b)$, where $0<b<a$ are reals. Let $\Gamma$ be a circle with diameter $\overline{AB}$ and let $P$ be any other point on $\Gamma$. Line $PA$ meets the x-axis again at $Q$. Prove that angle $\angle BQP = \angle BOP$.
2017 BAMO, 3
Consider the $n \times n$ “multiplication table” below. The numbers in the first column multiplied by the numbers in the first row give the remaining numbers in the table.
[asy]
import graph;
size(3.5cm);
for (int x=0; x<=5; ++x)
draw((x, 0) -- (x, 5), linewidth(.5pt));
for (int y=0; y<=5; ++y)
draw((0, y) -- (5, y), linewidth(.5pt));
draw((0,0)--(5,0)--(5,5)--(0,5)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
foo(0, 4, "1");
foo(1, 4, "2");
foo(2, 4, "3");
foo(3, 4, "$\dots$");
foo(4, 4, "$n$");
foo(0, 3, "2");
foo(1, 3, "4");
foo(2, 3, "6");
foo(3, 3, "$\dots$");
foo(4, 3, "$2n$");
foo(0, 2, "3");
foo(1, 2, "6");
foo(2, 2, "9");
foo(3, 2, "$\dots$");
foo(4, 2, "$3n$");
foo(0, 1, "$\vdots$");
foo(1, 1, "$\vdots$");
foo(2, 1, "$\vdots$");
foo(3, 1, "$\ddots$");
foo(4, 1, "$\vdots$");
foo(0, 0, "$n$");
foo(1, 0, "$2n$");
foo(2, 0, "$3n$");
foo(3, 0, "$\dots$");
foo(4, 0, "$n^2$");
[/asy]
We create a path from the upper-left square to the lower-right square by always moving one cell either to the right or down. For example, in the case $n = 5$, here is one such possible path, with all the numbers along the path circled:
[asy]
import graph;
size(3.5cm);
for (int x=0; x<=5; ++x)
draw((x, 0) -- (x, 5), linewidth(.5pt));
for (int y=0; y<=5; ++y)
draw((0, y) -- (5, y), linewidth(.5pt));
draw((0,0)--(5,0)--(5,5)--(0,5)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
draw(Circle((0.5,4.5),0.5));
draw(Circle((1.5,4.5),0.5));
draw(Circle((2.5,4.5),0.5));
draw(Circle((2.5,3.5),0.5));
draw(Circle((3.5,3.5),0.5));
draw(Circle((3.5,2.5),0.5));
draw(Circle((3.5,1.5),0.5));
draw(Circle((3.5,0.5),0.5));
draw(Circle((4.5,0.5),0.5));
foo(0, 4, "1");
foo(1, 4, "2");
foo(2, 4, "3");
foo(3, 4, "4");
foo(4, 4, "5");
foo(0, 3, "2");
foo(1, 3, "4");
foo(2, 3, "6");
foo(3, 3, "8");
foo(4, 3, "10");
foo(0, 2, "3");
foo(1, 2, "6");
foo(2, 2, "9");
foo(3, 2, "12");
foo(4, 2, "15");
foo(0, 1, "4");
foo(1, 1, "8");
foo(2, 1, "12");
foo(3, 1, "16");
foo(4, 1, "20");
foo(0, 0, "5");
foo(1, 0, "10");
foo(2, 0, "15");
foo(3, 0, "20");
foo(4, 0, "25");
[/asy]
If we add up the circled numbers in the example above (including the start and end squares), we get $93$. Considering all such possible paths on the $n \times n$ grid:
(a) What is the smallest sum we can possibly get when we add up the numbers along such a path? Express your answer in terms of $n$, and prove that it is correct.
(b) What is the largest sum we can possibly get when we add up the numbers along such a path? Express your answer in terms of $n$, and prove that it is correct.
2023 BAMO, 5
A [i]lattice point[/i] in the plane is a point with integer coordinates. Let $T$ be a triangle in the plane whose vertice are lattice points, but with no other lattice points on its sides. Furthermore, suppose $T$ contains exactly four lattice points in its interior. Prove that these four points lie on a straight line.
2022 BAMO, A
If I have 100 cards with all the numbers 1 through 100 on them, how should I put them in order to create the largest possible number?
2017 BAMO, B
Two three-dimensional objects are said to have the same coloring if you can orient one object (by moving or turning it) so that it is indistinguishable from the other. For example, suppose we have two unit cubes sitting on a table, and the faces of one cube are all black except for the top face which is red, and the the faces of the other cube are all black except for the bottom face, which is colored red. Then these two cubes have the same coloring.
In how many different ways can you color the edges of a regular tetrahedron, coloring two edges red, two edges black, and two edges green? (A regular tetrahedron has four faces that are each equilateral triangles. The figure below depicts one coloring of a tetrahedron, using thick, thin, and dashed lines to indicate three colors.)
2024 BAMO, E/3
Let $S_n$ be the sum of the first $n$ prime numbers. For example,
\[ S_5 = 2 + 3 + 5 + 7 + 11 = 28.\]
Does there exist an integer $k$ such that $S_{2023} < k^2 < S_{2024}$?
2017 BAMO, E/4
Consider a convex $n$-gon $A_1A_2 \dots A_n$. (Note: In a convex polygon, all interior angles are less than $180 \circ$.) Let $h$ be a positive number. Using the sides of the polygon as bases, we draw $n$ rectangles, each of height $h$, so that each rectangle is either entirely inside the $n$-gon or partially overlaps the inside of the $n$-gon.
As an example, the left figure below shows a pentagon with a correct configuration of rectangles, while the right figure shows an incorrect configuration of rectangles (since some of the rectangles do not overlap with the pentagon):
2022 BAMO, D/2
Suppose that $p,p+d,p+2d,p+3d,p+4d$, and $p+5d$ are six prime numbers, where $p$ and $d$ are positive integers. Show that $d$ must be divisible by $2,3,$ and $5$.
2024 BAMO, 4
Find all polynomials $f$ that satisfy the equation
\[\frac{f(3x)}{f(x)} = \frac{729 (x-3)}{x-243}\]
for infinitely many real values of $x$.
2017 BAMO, D/2
The area of square $ABCD$ is $196 \text{cm}^2$. Point $E$ is inside the square, at the same distances from points $D$ and $C$, and such that $m \angle DEC = 150^{\circ}$. What is the perimeter of $\triangle ABE$ equal to? Prove your answer is correct.
2024 BAMO, D/2
Sasha wants to bake $6$ cookies in his $8$ inch $\times$ $8$ inch square baking sheet. With a cookie cutter, he cuts out from the dough six circular shapes, each exactly $3$ inches in diameter. Can he place these six dough shapes on the baking sheet without the shapes touching each other? If yes, show us how. If no, explain why not. (Assume that the dough does not expand during baking.)
2024 BAMO, A
A school needs to elect its president. The school has $121$ students, each of whom belongs to one of two tribes: Geometers or Algebraists. Two candidates are running for president: one Geometer and one Algebraist. The Geometers vote only for Geometers and the Algebraists only for Algebraists. There are more Algebraists than Geometers, but the Geometers are resourceful. They convince the school that the following two-step procedure is fairer:
[list=a]
[*]The school is divided into $11$ groups, with $11$ students in each group. Each group elects a representative for step 2.
[*]The $11$ elected representatives elect a president.
[/list]
Not only do the Geometers manage to have this two-step procedure approved, they also volunteer to assign the students to groups for step 1. What is the minimum number of Geometers in the school that guarantees they can elect a Geometer as president? (In any stage of voting, the majority wins.)
2022 BAMO, 4
Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$-meter portion of the wire, there is an $\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?
2010 BAMO, 5
Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Prove that
$1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]$ is greater than or equal to $2^{1/2}$.
2017 BAMO, A
Consider the $4 \times 4$ “multiplication table” below. The numbers in the first column multiplied by the numbers in the first row give the remaining numbers in the table. For example, the $3$ in the first column times the $4$ in the first row give the $12 (= 3 \cdot 4)$ in the cell that is in the 3rd row and 4th column.
[asy]
size(3cm);
for (int x=0; x<=4; ++x)
draw((x, 0) -- (x, 4), linewidth(.5pt));
for (int y=0; y<=4; ++y)
draw((0, y) -- (4, y), linewidth(.5pt));
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
foo(0, 3, "1");
foo(1, 3, "2");
foo(2, 3, "3");
foo(3, 3, "4");
foo(0, 2, "2");
foo(1, 2, "4");
foo(2, 2, "6");
foo(3, 2, "8");
foo(0, 1, "3");
foo(1, 1, "6");
foo(2, 1, "9");
foo(3, 1, "12");
foo(0, 0, "4");
foo(1, 0, "8");
foo(2, 0, "12");
foo(3, 0, "16");
[/asy]
We create a path from the upper-left square to the lower-right square by always moving one cell either to the right or down. For example, here is one such possible path, with all the numbers along the path circled:
[asy]
import graph;
size(3cm);
for (int x=0; x<=4; ++x)
draw((x, 0) -- (x, 4), linewidth(.5pt));
for (int y=0; y<=4; ++y)
draw((0, y) -- (4, y), linewidth(.5pt));
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
draw(Circle((0.5,3.5),0.5));
draw(Circle((1.5,3.5),0.5));
draw(Circle((2.5,3.5),0.5));
draw(Circle((2.5,2.5),0.5));
draw(Circle((3.5,2.5),0.5));
draw(Circle((3.5,1.5),0.5));
draw(Circle((3.5,0.5),0.5));
foo(0, 3, "1");
foo(1, 3, "2");
foo(2, 3, "3");
foo(3, 3, "4");
foo(0, 2, "2");
foo(1, 2, "4");
foo(2, 2, "6");
foo(3, 2, "8");
foo(0, 1, "3");
foo(1, 1, "6");
foo(2, 1, "9");
foo(3, 1, "12");
foo(0, 0, "4");
foo(1, 0, "8");
foo(2, 0, "12");
foo(3, 0, "16");
[/asy]
If we add up the circled numbers in the example above (including the start and end squares), we get $48$. Considering all such possible paths:
(a) What is the smallest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.
(b) What is the largest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.
2023 BAMO, D/2
Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.)
What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)
2022 BAMO, B
You are bargaining with a salesperson for the price of an item. Your first offer is $a$ dollars and theirs is $b$ dollars. After you raise your offer by a certain percentage and they lower their offer by the same percentage, you arrive at an agreed price. What is that price, in terms of $a$ and $b$?
2022 BAMO, 5
Sofiya and Marquis are playing a game. Sofiya announces to Marquis that she's thinking of a polynomial of the form $f(x)=x^3+px+q$ with three integer roots that are not necessarily distinct. She also explains that all of the integer roots have absolute value less than (and not equal to) $N$, where $N$ is some fixed number which she tells Marquis. As a "move" in this game, Marquis can ask Sofiya about any number $x$ and Sofiya will tell him whether $f(x)$ is positive negative, or zero. Marquis's goal is to figure out Sofiya's polynomial.
If $N=3\cdot 2^k$ for some positive integer $k$, prove that there is a strategy which allows Marquis to identify the polynomial after making at most $2k+1$ "moves".
2020 BAMO, D/2
Here’s a screenshot of the problem. If someone could LaTEX a diagram, that would be great!
2017 BAMO, 5
Call a number $T$ [i]persistent[/i] if the following holds: Whenever $a,b,c,d$ are real numbers different from $0$ and $1$ such that
$$a+b+c+d = T$$
and
$$\frac{1}{a}+\frac{1}{b} +\frac{1}{c}+\frac{1}{d} = T,$$
we also have
$$\frac{1}{1 - a}+\frac{1}{1-b}+\frac{1}{1-c}+\frac{1}{1-d}= T.$$
(a) If $T$ is persistent, prove that $T$ must be equal to $2$.
(b) Prove that $2$ is persistent.
Note: alternatively, we can just ask “Show that there exists a unique persistent number, and determine its value”.
2022 BAMO, C/1
The game of pool includes $15$ balls that fit within a triangular rack as shown:
[asy]
// thanks Ritwin for this diagram :D
unitsize(0.6cm);
pair pos(real i, real j) {
return i*dir(60) + (j,0);
}
for (int i = 0; i <= 4; ++i) {
for (int j = 0; j <= 4-i; ++j) {
draw(circle(pos(i,j), .5));
}
}
pair A = pos(0,0);
pair B = pos(0,4);
pair C = pos(4,0);
pair dd = dir(270) * .5;
pair ul = dir(150) * .5;
pair ur = dir( 30) * .5;
real S = 1.75;
draw(A+dd -- B+dd ^^ B+ur -- C+ur ^^ C+ul -- A+ul );
draw(A+dd*S -- B+dd*S ^^ B+ur*S -- C+ur*S ^^ C+ul*S -- A+ul*S);
draw(arc(A, A+ul*S, A+dd*S));
draw(arc(B, B+dd*S, B+ur*S));
draw(arc(C, C+ur*S, C+ul*S));
[/asy]
Seven of the balls are "striped" (not colored with a single color) and eight are "solid" (colored with a single color). Prove that no matter how the $15$ balls are arranged in the rack, there must always be a pair of striped balls adjacent to each other.
2024 BAMO, 5
An underground burrow consists of an infinite sequence of rooms labeled by the integers $(\dots, -3, -2, -1, 0, 1, 2, 3,\dots)$. Initially, some of the rooms are occupied by one or more rabbits. Each rabbit wants to be alone. Thus, if there are two or more rabbits in the same room (say, room $m$), half of the rabbits (rounding down) will flee to room $m-1$, and half (also rounding down) to room $m+1$. Once per minute, this happens simultaneously in all rooms that have two or more rabbits. For example, if initially all rooms are empty except for $5$ rabbits in room $\#12$ and $2$ rabbits in room $\#13$, then after one minute, rooms $\text{\#11--\#14}$ will contain $2$, $2$, $2$, and $1$ rabbits, respectively, and all other rooms will be empty.
Now suppose that initially there are $k+1$ rabbits in room $k$ for each $k=0, 1, 2, \ldots, 9, 10$, and all other rooms are empty.
[list=a]
[*]Show that eventually the rabbits will stop moving.
[*] Determine which rooms will be occupied when this occurs.
[/list]
2024 BAMO, C/1
Sugar Station sells $44$ different kinds of candies, packaged one to a box. Each box is priced at a positive integer number of cents, and it costs $\$1.51$ to buy one of every kind. (There is no discount based on the number of candies in a purchase.) Unfortunately, Anna only has $\$0.75$.
[list=a]
[*] Show that Anna can buy at least $22$ boxes, each containing a different candy.
[*] Show that Anna can do even better, buying at least $25$ boxes, each containing a different candy.
[/list]