Found problems: 85335
2019 Saudi Arabia JBMO TST, 1
All points in the plane are colored in $n$ colors. In each line, there are point of no more than two colors. What is the maximum number of colors?
2024 CCA Math Bonanza, L2.2
Let a rad number be a palindrome such that the square root of the sum of its digits is irrational. Find the number of $4$-digit rad numbers.
[i]Lightning 2.2[/i]
2010 USAJMO, 6
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.
2000 Austria Beginners' Competition, 1
Let $a$ be a real number. Determine, for all $a$, all pairs $(x,y)$ of real numbers such that $(x-y^2)(y-x^2)+x^3+y^3=a $.
1977 AMC 12/AHSME, 18
If $y=(\log_23)(\log_34)\cdots(\log_n[n+1])\cdots(\log_{31}32)$ then
$\textbf{(A) }4<y<5\qquad\textbf{(B) }y=5\qquad\textbf{(C) }5<y<6\qquad$
$\textbf{(D) }y=6\qquad \textbf{(E) }6<y<7$
2021 Iran Team Selection Test, 1
Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.
(Two cells of the grid are adjacent if they have a common vertex)
1999 AMC 12/AHSME, 2
Which of the following statements is false?
$ \textbf{(A)}\ \text{All equilateral triangles are congruent to each other.}$
$ \textbf{(B)}\ \text{All equilateral triangles are convex.}$
$ \textbf{(C)}\ \text{All equilateral triangles are equilangular.}$
$ \textbf{(D)}\ \text{All equilateral triangles are regular polygons.}$
$ \textbf{(E)}\ \text{All equilateral triangles are similar to each other.}$
2016 CMIMC, 5
Let $\mathcal{S}$ be a regular 18-gon, and for two vertices in $\mathcal{S}$ define the $\textit{distance}$ between them to be the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g. adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distance 1, 8, or 9.
2012 China National Olympiad, 3
Prove for any $M>2$, there exists an increasing sequence of positive integers $a_1<a_2<\ldots $ satisfying:
1) $a_i>M^i$ for any $i$;
2) There exists a positive integer $m$ and $b_1,b_2,\ldots ,b_m\in\left\{ -1,1\right\}$, satisfying $n=a_1b_1+a_2b_2+\ldots +a_mb_m$ if and only if $n\in\mathbb{Z}/ \{0\}$.
2021 AMC 10 Fall, 17
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at $A$, $B$, and $C$ are $12,9,$ and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$?
$\textbf{(A) }9\qquad\textbf{(B) }6\sqrt3\qquad\textbf{(C) }8\sqrt3\qquad\textbf{(D) }17\qquad\textbf{(E) }12\sqrt3$
2015 Geolympiad Summer, 1.
Show in an acute triangle $ABC$ that $\cot A + \cot B + \cot C \ge \dfrac{12[ABC]}{a^2+b^2+c^2}$.
2016 IOM, 3
Let $A_1A_2 . . . A_n$ be a cyclic convex polygon whose circumcenter is strictly in its interior. Let $B_1, B_2, ..., B_n$ be arbitrary points on the sides $A_1A_2, A_2A_3, ..., A_nA_1$, respectively, other than the vertices. Prove that
$\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+...+\frac{B_nB_1}{A_nA_2}>1$.
2016 Online Math Open Problems, 11
Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9)>f(16)$. The probability that $f(1)>f(16)>f(25)$ can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1,2, \ldots, 100\}$.
[i]Proposed by Evan Chen[/i]
2024 Bangladesh Mathematical Olympiad, P5
Let $I$ be the incenter of $\triangle ABC$ and $P$ be a point such that $PI$ is perpendicular to $BC$ and $PA$ is parallel to $BC$. Let the line parallel to $BC$, which is tangent to the incircle of $\triangle ABC$, intersect $AB$ and $AC$ at points $Q$ and $R$ respectively. Prove that $\angle BPQ = \angle CPR$.
2003 May Olympiad, 5
We have a $4 \times 4$ squared board. We define the [i]separation [/i] between two squares as the least number of moves that a chess knight must take to go from one square to the other (using moves of the knight). Three boxes $A, B, C$ form a good trio if the three separations between $A$ and $B$, between $A$ and $C$ and between $B$ and $C$ are equal. Determines the number of good trios that are formed on the board.
Clarification: In each move the knight moves $2$ squares in the horizontal direction plus one square in the vertical direction or moves $2$ squares in the vertical direction plus one square in the horizontal direction.
1967 IMO Longlists, 42
Decompose the expression into real factors:
\[E = 1 - \sin^5(x) - \cos^5(x).\]
2023 Czech-Polish-Slovak Match, 2
Let $a_1, a_2, \ldots, a_n$ be reals such that for all $k=1,2, \ldots, n$, $na_k \geq a_1^2+a_2^2+ \ldots+a_k^2$. Prove that there exist at least $\frac{n} {10}$ indices $k$, such that $a_k \leq 1000$.
2015 Hanoi Open Mathematics Competitions, 3
Suppose that $a > b > c > 1$. One of solutions of the equation
$\frac{(x - a)(x - b)}{(c - a)(c - b)}+\frac{(x - b)(x - c)}{(a - b)(a - c)}+\frac{(x - c)(x - a)}{(b - c)(b - a)}= x$ is
(A): $-1$, (B): $-2$, (C): $0$, (D): $1$, (E): None of the above.
2016 International Zhautykov Olympiad, 2
A convex hexagon $ABCDEF$ is given such that $AB||DE$, $BC||EF$, $CD||FA$. The point $M, N, K$ are common points of the lines $BD$ and $AE$, $AC$ and $DF$, $CE$ and $BF$ respectively. Prove that perpendiculars drawn from $M, N, K$ to lines $AB, CD, EF$ respectively concurrent.
1979 IMO Shortlist, 23
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.
2020 AMC 12/AHSME, 9
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
[asy]
draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));
label("$4$", (2,0), S);
[/asy]
$\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$
2022 Bulgarian Autumn Math Competition, Problem 11.4
The number $2022$ is written on the white board. Ivan and Peter play a game, Ivan starts and they alternate. On a move, Ivan erases the number $b$, written on the board, throws a dice which shows some number $a$, and writes the residue of $(a+b) ^2$ modulo $5$. Similarly, Peter throws a dice which shows some number $a$, and changes the previously written number $b$ to the residue of $a+b$ modulo $3$. The first player to write a $0$ wins. What is the probability of Ivan winning the game?
2010 Contests, 1b
The edges of the square in the figure have length $1$.
Find the area of the marked region in terms of $a$, where $0 \le a \le 1$.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]
2003 AMC 12-AHSME, 20
Part of the graph of $ f(x) \equal{} x^3 \plus{} bx^2 \plus{} cx \plus{} d$ is shown. What is $ b$?
[asy]import graph;
unitsize(1.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;
real y(real x)
{
return (x-1)*(x+1)*(x-2);
}
path bounds=(-1.5,-1)--(1.5,-1)--(1.5,2.5)--(-1.5,2.5)--cycle;
pair[] points={(-1,0),(0,2),(1,0)};
draw(bounds,white);
draw(graph(y,-1.5,1.5));
drawline((0,0),(1,0));
drawline((0,0),(0,1));
dot(points);
label("$(-1,0)$",(-1,0),SE);
label("$(1,0)$",(1,0),SW);
label("$(0,2)$",(0,2),NE);
clip(bounds);[/asy]$ \textbf{(A)}\minus{}\!4 \qquad \textbf{(B)}\minus{}\!2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 4$
2013 Canadian Mathematical Olympiad Qualification Repechage, 3
A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.