Found problems: 85335
JOM 2023, 2
Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$)
[i]Proposed by Wong Jer Ren[/i]
1986 Traian Lălescu, 1.3
Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.
2022 New Zealand MO, 8
Find all real numbers $x$ such that $-1 < x \le 2 $ and
$$\sqrt{2 - x}+\sqrt{2 + 2x} =\sqrt{\frac{x^4 + 1}{x^2 + 1}}+ \frac{x + 3}{x + 1}.$$
.
2003 Mexico National Olympiad, 1
Find all positive integers with two or more digits such that if we insert a $0$ between the units and tens digits we get a multiple of the original number.
2001 239 Open Mathematical Olympiad, 2
In a convex quadrangle $ ABCD $, the rays $ DA $ and $ CB $ intersect at point $ Q $, and the rays $ BA $ and $ CD $ at the point $ P $. It turned out that $ \angle AQB = \angle APD $. The bisectors of the angles $ \angle AQB $ and $ \angle APD $ intersect the sides quadrangle at points $ X $, $ Y $ and $ Z $, $ T $ respectively. Circumscribed circles of triangles $ ZQT $ and $ XPY $ intersect at $ K $ inside quadrangle. Prove that $ K $ lies on the diagonal $ AC $.
2001 AMC 10, 4
What is the maximum number of possible points of intersection of a circle and a triangle?
$ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
2006 German National Olympiad, 2
Five points are on the surface of of a sphere of radius $1$. Let $a_{\text{min}}$ denote the smallest distance (measured along a straight line in space) between any two of these points. What is the maximum value for $a_{\text{min}}$, taken over all arrangements of the five points?
2014 Czech and Slovak Olympiad III A, 1
Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$.
Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$.
(Day 1, 1st problem
author: Matúš Harminc)
2022 IFYM, Sozopol, 8
A magician wants to demonstrate the following trick to an audience of $n \ge 16$ people. He gives them $15$ hats and after giving instructions to his assistant (which the audience does not hear), leaves the hall. Some $15$ people in the audience put on one of the hats. The assistant tags in front of everyone, one of the hats with a marker and then the person with an unmarked hat takes it off. The magician then returns back to the hall and after surveying the situation, knows who in the audience has taken off his hat. For what $n$ is this possible?
[hide=original wording]Магьосник иска да покаже следния фокус пред публика от $n \ge 16$ души. Той им дава $15$ шапки и след като даде инструкции на помощника си (които публиката не чува), напуска залата. Някои $15$ души от публиката си слагат по една от шапките. Асистентът маркира пред всички една от шапките с маркер и след това човек с немаркирана шапка си я сваля. След това магьосникът се връща обратно в залата и след оглед на ситуацията познава кой от публиката си е свалил шапката. За кои $n$ е възможно това?[/hide]
2005 Harvard-MIT Mathematics Tournament, 1
The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?
1962 Putnam, B1
Let $x^{(n)}=x(x-1)\cdots (x-n+1)$ for $n$ a positive integer and let $x^{(0)}=1.$ Prove that
$$(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.$$
2020 Thailand TSTST, 1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through point $A$, meets segments $AB$ and $AC$ again at $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $(BDF)$ at $F$ and the tangent to circle $(CEG)$ at $G$ meet at $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
2017 AIME Problems, 10
Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\overline{AD}$, point $N$ is the trisection point of $\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\overline{CM}$ and $\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\overline{BP}$ bisects the area of $BCON$. Find the area of $\triangle{CDP}$.
2019 China Team Selection Test, 6
Given positive integers $d \ge 3$, $r>2$ and $l$, with $2d \le l <rd$. Every vertice of the graph $G(V,E)$ is assigned to a positive integer in $\{1,2,\cdots,l\}$, such that for any two consecutive vertices in the graph, the integers they are assigned to, respectively, have difference no less than $d$, and no more than $l-d$.
A proper coloring of the graph is a coloring of the vertices, such that any two consecutive vertices are not the same color. It's given that there exist a proper subset $A$ of $V$, such that for $G$'s any proper coloring with $r-1$ colors, and for an arbitrary color $C$, either all numbers in color $C$ appear in $A$, or none of the numbers in color $C$ appear in $A$.
Show that $G$ has a proper coloring within $r-1$ colors.
Putnam 1938, B6
What is the shortest distance between the plane $Ax + By + Cz + 1 = 0$ and the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$ You may find it convenient to use the notation $h = (A^2 + B^2 + C^2)^{\frac{-1}{2}}, m = (a^2A^2 + b^2B^2 + c^2C^2)^{\frac{1}{2}}.$ What is the algebraic condition for the plane not to intersect the ellipsoid?
2020 AMC 12/AHSME, 24
Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$
$\textbf{(A) } 1 + \sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$
2024 Lusophon Mathematical Olympiad, 1
Determine all geometric progressions such that the product of the three first terms is $64$ and the sum of them is $14$.
1959 Putnam, A4
If $f$ and $g$ are real-valued functions of one real variable, show that there exist $x$ and $y$ in $[0,1]$ such that $$|xy-f(x)-g(y)|\geq \frac{1}{4}.$$
2013 Stanford Mathematics Tournament, 9
Charles is playing a variant of Sudoku. To each lattice point $(x, y)$ where $1\le x,y <100$, he assigns an integer between $1$ and $100$ inclusive. These integers satisfy the property that in any row where $y=k$, the $99$ values are distinct and never equal to $k$; similarly for any column where $x=k$. Now, Charles randomly selects one of his lattice points with probability proportional to the integer value he assigned to it. Compute the expected value of $x+y$ for the chosen point $(x, y)$.
2012 AMC 12/AHSME, 25
Let $S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}$, and $(x,y) \neq (0,0) \}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle ABC$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)= \tan (\angle CBA)$. What is
\[ \displaystyle \prod_{t \in T} f(t) \text{?} \]
[asy]
size((120));
dot((1,0));
dot((2,0));
dot((3,0));
dot((4,0));
dot((0,1));
dot((0,2));
dot((0,3));
dot((0,4));
dot((0,5));
dot((1,1));
dot((1,2));
dot((1,3));
dot((1,4));
dot((1,5));
dot((2,1));
dot((2,2));
dot((2,3));
dot((2,4));
dot((2,5));
dot((3,1));
dot((3,2));
dot((3,3));
dot((3,4));
dot((3,5));
dot((4,1));
dot((4,2));
dot((4,3));
dot((4,4));
dot((4,5));
label("$\circ$", (0,0));
label("$S$", (-.7,2.5));
[/asy]
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{625}{144} \qquad \textbf{(C)}\ \frac{125}{24} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{625}{24}$
2010 Contests, 3
In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$
such that $BC$ not is the diameter.Consider a point $A$ varies in
$(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$
respective is intersect of $BC$ and internal and external bisector
of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through
orthocenter of $\triangle ABC$
and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.
1/Prove that $MN$ pass through a fixed point
2/Determint the place of $A$ such that $S_{AMN}$ has maxium value
2023 LMT Fall, 6
Jeff rolls a standard $6$ sided die repeatedly until he rolls either all of the prime numbers possible at least once, or all the of even numbers possible at least once. Find the probability that his last roll is a $2$.
1982 Yugoslav Team Selection Test, Problem 3
Let there be given real numbers $x_i>1~(i=1,2,\ldots,2n)$. Prove that the interval $[0,2]$ contains at most $\binom{2n}n$ sums of the form $\alpha_1x_1+\ldots+\alpha_{2n}x_{2n}$, where $\alpha_i\in\{-1,1\}$ for all $i$.
2022 USA TSTST, 1
Let $n$ be a positive integer. Find the smallest positive integer $k$ such that for any set $S$ of $n$ points in the interior of the unit square, there exists a set of $k$ rectangles such that the following hold:
[list=disc]
[*]The sides of each rectangle are parallel to the sides of the unit square.
[*]Each point in $S$ is [i]not[/i] in the interior of any rectangle.
[*]Each point in the interior of the unit square but [i]not[/i] in $S$ is in the interior of at least one of the $k$ rectangles
[/list]
(The interior of a polygon does not contain its boundary.)
[i]Holden Mui[/i]
2022 Kyiv City MO Round 1, Problem 4
Let's call integer square-free if it's not divisible by $p^2$ for any prime $p$. You are given a square-free integer $n>1$, which has exactly $d$ positive divisors. Find the largest number of its divisors that you can choose, such that $a^2 + ab - n$ isn't a square of an integer for any $a, b$ among chosen divisors.
[i](Proposed by Oleksii Masalitin)[/i]