Found problems: 85335
1998 Iran MO (3rd Round), 2
Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that
\[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC.
\]
Prove that
\[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1.
\]
2006 Romania National Olympiad, 3
Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.
2013 USAMTS Problems, 5
Let $S$ be a planar region. A $\emph{domino-tiling}$ of $S$ is a partition of $S$ into $1\times2$ rectangles. (For example, a $2\times3$ rectangle has exactly $3$ domino-tilings, as shown below.)
[asy]
import graph; size(7cm);
pen dps = linewidth(0.7); defaultpen(dps);
draw((0,0)--(3,0)--(3,2)--(0,2)--cycle, linewidth(2));
draw((4,0)--(4,2)--(7,2)--(7,0)--cycle, linewidth(2));
draw((8,0)--(8,2)--(11,2)--(11,0)--cycle, linewidth(2));
draw((1,0)--(1,2));
draw((2,1)--(3,1));
draw((0,1)--(2,1), linewidth(2));
draw((2,0)--(2,2), linewidth(2));
draw((4,1)--(7,1));
draw((5,0)--(5,2), linewidth(2));
draw((6,0)--(6,2), linewidth(2));
draw((8,1)--(9,1));
draw((10,0)--(10,2));
draw((9,0)--(9,2), linewidth(2));
draw((9,1)--(11,1), linewidth(2));
[/asy]
The rectangles in the partition of $S$ are called $\emph{dominoes}$.
(a) For any given positive integer $n$, find a region $S_n$ with area at most $2n$ that has exactly $n$ domino-tilings.
(b) Find a region $T$ with area less than $50000$ that has exactly $100002013$ domino-tilings.
1998 North Macedonia National Olympiad, 1
Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$.
Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$
2014-2015 SDML (Middle School), 7
Gizmo is thinking of a geometric sequence in which the third term is $1215$ and the fifth is $540$. Which of the following could be the eighth term of Gizmo's sequence?
$\text{(A) }-160\qquad\text{(B) }-135.5\qquad\text{(C) }216\qquad\text{(D) }240\qquad\text{(E) }472.5$
1988 National High School Mathematics League, 1
Define sequence $(a_n):a_1=1,a_2=2,a_{n+2}=\begin{cases}
5a_{n+1}-3a_n,\text{if }a_n\cdot a_{n+1}\text{ is even}\\
a_{n+1}-a_n,\text{if }a_n\cdot a_{n+1}\text{ is odd}
\end{cases}$
Prove that for all $n\in\mathbb{Z}_+$, $a_n\neq0$.
1976 Chisinau City MO, 124
Find $3$ numbers, each of which is equal to the square of the difference of the other two.
2011 Hanoi Open Mathematics Competitions, 5
Let M = 7!.8!.9!.10!.11!.12!. How many factors of M are perfect squares ?
1962 IMO Shortlist, 3
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
1983 AMC 12/AHSME, 29
A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A$, $B$, $C$ and $D$. Also, let the distances from $P$ to $A$, $B$ and $C$, respectively, be $u$, $v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$?
$ \textbf{(A)}\ 1 + \sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{2}\qquad\textbf{(C)}\ 2 + \sqrt{2}\qquad\textbf{(D)}\ 3\sqrt{2}\qquad\textbf{(E)}\ 3 + \sqrt{2}$
2004 Estonia National Olympiad, 3
On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.
Cono Sur Shortlist - geometry, 2009.G4
Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.
1999 All-Russian Olympiad Regional Round, 8.1
A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours
2021-2022 OMMC, 12
Katelyn is building an integer (in base $10$). She begins with $9$. Each step, she appends a randomly chosen digit from $0$ to $9$ inclusive to the right end of her current integer. She stops immediately when the current integer is $0$ or $1$ (mod $11$). The probability that the final integer ends up being $0$ (mod $11$) is $\tfrac ab$ for coprime positive integers $a$, $b$. Find $a + b$.
[i]Proposed by Evan Chang[/i]
2014 China Western Mathematical Olympiad, 7
In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$.
Prove that\[|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.\]
2004 Romania National Olympiad, 2
Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$.
(a) Prove that $f_n$ is well defined.
(b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective.
[i]Bogdan Enescu[/i]
Today's calculation of integrals, 869
Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$
Answer the questions below.
(1) Find $I_n.$
(2) Find $\sum_{n=1}^{\infty} I_n.$
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
1953 AMC 12/AHSME, 31
The rails on a railroad are $ 30$ feet long. As the train passes over the point where the rails are joined, there is an audible click. The speed of the train in miles per hour is approximately the number of clicks heard in:
$ \textbf{(A)}\ 20\text{ seconds} \qquad\textbf{(B)}\ 2\text{ minutes} \qquad\textbf{(C)}\ 1\frac{1}{2}\text{ minutes} \qquad\textbf{(D)}\ 5\text{ minutes}\\
\textbf{(E)}\ \text{none of these}$
2014-2015 SDML (High School), 1
If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?
2002 Estonia Team Selection Test, 1
The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.
2016 AIME Problems, 11
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.
2010 Putnam, A5
Let $G$ be a group, with operation $*$. Suppose that
(i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);
(ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$
Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$
2018 ELMO Shortlist, 3
Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that \[a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.\]
[i]Proposed by Daniel Liu[/i]
2015 Chile TST Ibero, 4
Let $x, y \in \mathbb{R}^+$. Prove that:
\[
\left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2.
\]