Found problems: 85335
2016 Taiwan TST Round 2, 1
In Lineland there are $n\geq1$ towns, arranged along a road running from left to right. Each town has a [i]left bulldozer[/i] (put to the left of the town and facing left) and a [i]right bulldozer[/i] (put to the right of the town and facing right). The sizes of the $2n$ bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes.
Let $A$ and $B$ be two towns, with $B$ to the right of $A$. We say that town $A$ can [i]sweep[/i] town $B$ [i]away[/i] if the right bulldozer of $A$ can move over to $B$ pushing off all bulldozers it meets. Similarly town $B$ can sweep town $A$ away if the left bulldozer of $B$ can move over to $A$ pushing off all bulldozers of all towns on its way.
Prove that there is exactly one town that cannot be swept away by any other one.
PEN P Problems, 28
Prove that any positive integer can be represented as a sum of Fibonacci numbers, no two of which are consecutive.
2006 China Team Selection Test, 2
Given three positive real numbers $ x$, $ y$, $ z$ such that $ x \plus{} y \plus{} z \equal{} 1$, prove that
$ \frac {xy}{\sqrt {xy \plus{} yz}} \plus{} \frac {yz}{\sqrt {yz \plus{} zx}} \plus{} \frac {zx}{\sqrt {zx \plus{} xy}} \le \frac {\sqrt {2}}{2}$.
2022 AMC 12/AHSME, 20
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$
2011 Junior Balkan Team Selection Tests - Romania, 3
a) Find the largest possible value of the number $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$, if $x_1, x_2, ... , x_n$ ($n \ge 2$) are non-negative integers and their sum is $2011$.
b) Find the numbers $x_1, x_2, ... , x_n$ for which the maximum value determined at a) is obtained
2025 Harvard-MIT Mathematics Tournament, 5
Compute the largest possible radius of a circle contained in the region defined by $|x+|y|| \le 1$ in the coordinate plane.
2015 Costa Rica - Final Round, N4
Show that there are no triples $(a, b, c)$ of positive integers such that
a) $a + c, b + c, a + b$ do not have common multiples in pairs.
b)$\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}$ are integer numbers.
2008 ITest, 12
One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is $3$ higher than Tony's. "Let's play once more," says Tony. Let $a/b$ be the probability that the difference between the outcomes of the two dice is again exactly $3$ (regardless of which of the boys rolls higher), where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
1990 Putnam, A6
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? Prove your answer.
2004 Romania National Olympiad, 1
Find all non-negative integers $n$ such that there are $a,b \in \mathbb Z$ satisfying $n^2=a+b$ and $n^3=a^2+b^2$.
[i]Lucian Dragomir[/i]
2019 All-Russian Olympiad, 8
A positive integer $n$ is given. A cube $3\times3\times3$ is built from $26$ white and $1$ black cubes $1\times1\times1$ such that the black cube is in the center of $3\times3\times3$-cube. A cube $3n\times 3n\times 3n$ is formed by $n^3$ such $3\times3\times3$-cubes. What is the smallest number of white cubes which should be colored in red in such a way that every white cube will have at least one common vertex with a red one.
[hide=thanks] Thanks to the user Vlados021 for translating the problem.[/hide]
1990 India Regional Mathematical Olympiad, 8
If the circumcenter and centroid of a triangle coincide, prove that it must be equilateral.
PEN F Problems, 8
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
2023 All-Russian Olympiad Regional Round, 9.9
Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.
2006 IMC, 4
Let $v_{0}$ be the zero ector and let $v_{1},...,v_{n+1}\in\mathbb{R}^{n}$ such that the Euclidian norm $|v_{i}-v_{j}|$ is rational for all $0\le i,j\le n+1$. Prove that $v_{1},...,v_{n+1}$ are linearly dependent over the rationals.
2017 China National Olympiad, 3
Consider a rectangle $R$ partitioned into $2016$ smaller rectangles such that the sides of each smaller rectangle is parallel to one of the sides of the original rectangle. Call the corners of each rectangle a vertex. For any segment joining two vertices, call it basic if no other vertex lie on it. (The segments must be part of the partitioning.) Find the maximum/minimum possible number of basic segments over all possible partitions of $R$.
2005 IMAR Test, 1
Let $a,b,c$ be positive real numbers such that $abc\geq 1$. Prove that \[ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. \]
[hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5[/url].
[/hide]
Kvant 2022, M2701
The king assembled 300 wizards and gave them the following challenge. For this challenge, 25 colors can be used, and they are known to the wizards. Each of the wizards receives a hat of one of those 25 colors. If for each color the number of used hats would be written down then all these number would be different, and the wizards know this. Each wizard sees what hat was given to each other wizard but does not see his own hat. Simultaneously each wizard reports the color of his own hat. Is it possible for the wizards to coordinate their actions beforehand so that at least 150 of them would report correctly?
2024 UMD Math Competition Part II, #4
Prove for every positive integer $n{:}$
\[ \frac {1 \cdot 3 \cdots (2n - 1)}{2 \cdot 4 \cdots (2n)} < \frac 1{\sqrt{3n}}\]
2012-2013 SDML (High School), 4
Circle $\omega_1$ with center $O_1$ has radius $3$, and circle $\omega_2$ with center $O_2$ has radius $2$ and is internally tangent to $\omega_1$. The segment $AB$ is a chord of $\omega_1$ that is tangent to $\omega_2$ at $C$ with $\angle{O_1O_2C}=45^{\circ}$. Find the length of $AB$.
[asy]
pair O_1, O_2, A, B, C;
O_1 = origin;
O_2 = (-1,0);
A = (-1, 2.82842712475);
B = (2.82842712475,-1);
C = O_2+2*dir(45);
dot(O_1);
dot(O_2);
dot(A);
dot(B);
dot(C);
draw(circle(O_1,3));
draw(circle(O_2,2));
draw(O_1--O_2);
draw(O_2--C);
draw(A--B);
label("$O_1$",O_1,SE);
label("$O_2$",O_2,SW);
label("$A$",A,NW);
label("$B$",B,SE);
label("$C$",C,NE);
[/asy]
2019 JBMO Shortlist, N2
Find all triples $(p, q, r)$ of prime numbers such that all of the following numbers are
integers
$\frac{p^2 + 2q}{q+r}, \frac{q^2+9r}{r+p}, \frac{r^2+3p}{p+q}$
[i]Proposed by Tajikistan[/i]
1983 AMC 12/AHSME, 23
In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$ ($L_1$ is the line that is above the circles and $L_2$ is the line that goes under the circles). If the radius of the largest circle is 18 and that of the smallest one is 8, then the radius of the middle circle is
[asy]
size(250);defaultpen(linewidth(0.7));
real alpha=5.797939254, x=71.191836;
int i;
for(i=0; i<5; i=i+1) {
real r=8*(sqrt(6)/2)^i;
draw(Circle((x+r)*dir(alpha), r));
x=x+2r;
}
real x=71.191836+40+20*sqrt(6), r=18;
pair A=tangent(origin, (x+r)*dir(alpha), r, 1), B=tangent(origin, (x+r)*dir(alpha), r, 2);
pair A1=300*dir(origin--A), B1=300*dir(origin--B);
draw(B1--origin--A1);
pair X=(69,-5), X1=reflect(origin, (x+r)*dir(alpha))*X,
Y=(200,-5), Y1=reflect(origin, (x+r)*dir(alpha))*Y,
Z=(130,0), Z1=reflect(origin, (x+r)*dir(alpha))*Z;
clip(X--Y--Y1--X1--cycle);
label("$L_2$", Z, S);
label("$L_1$", Z1, dir(2*alpha)*dir(90));[/asy]
$\text{(A)} \ 12 \qquad \text{(B)} \ 12.5 \qquad \text{(C)} \ 13 \qquad \text{(D)} \ 13.5 \qquad \text{(E)} \ 14$
OMMC POTM, 2022 11
Let $S$ be the set of colorings of a $100 \times 100$ grid where each square is colored black or white and no $2\times2$ subgrid is colored like a chessboard. A random such coloring is chosen: what is the probability there is a path of black squares going from the top row to the bottom row where any two consecutive squares in the path are adjacent?
[i]Proposed by Evan Chang (squareman), USA [/i]
2019 AMC 12/AHSME, 19
In $\triangle ABC$ with integer side lengths,
\[
\cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}.
\] What is the least possible perimeter for $\triangle ABC$?
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 44$
1989 China Team Selection Test, 4
$\forall n \in \mathbb{N}$, $P(n)$ denotes the number of the partition of $n$ as the sum of positive integers (disregarding the order of the parts), e.g. since $4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4$, so $P(4)=5$. "Dispersion" of a partition denotes the number of different parts in that partitation. And denote $q(n)$ is the sum of all the dispersions, e.g. $q(4)=1+2+2+1+1=7$. $n \geq 1$. Prove that
(1) $q(n) = 1 + \sum^{n-1}_{i=1} P(i).$
(2) $1 + \sum^{n-1}_{i=1} P(i) \leq \sqrt{2} \cdot n \cdot P(n)$.