$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.

What is the sum of all numbers between $0$ and $511$ inclusive that have an even number of $1$s when written in binary?

Construct a triangle by its altitude , median and angle bisector originating from one vertex.

Point $O$ is selected in equilateral $\triangle ABC$ such that the sum of the distances from $O$ to each side of $ABC$ is $15$. Compute the area of $ABC$. [center][img]https://cdn.artofproblemsolving.com/attachments/4/0/dd573985a7c98f23fd05d11e95c4b908eaa895.png[/img][/center] $\textbf{(A) } 15\sqrt3\qquad\textbf{(B) } 30\sqrt3\qquad\textbf{(C) } 50\sqrt3\qquad\textbf{(D) } 75\sqrt3\qquad\textbf{(E) } 225\sqrt3$

Given a line $p$ and a triangle $\Delta$ in the plane, construct an equilateral triangle one of whose vertices lies on the line $p$, while the other two halve the perimeter of $\Delta.$

Consider a $3\times7$ grid of squares. Each square may be coloured green or white. [list] (a) Is it possible to find a colouring so that no subrectangle has all four corner squares of the same colour? (b) Is it possible for a $4\times 6$ grid? [/list] [i]Subrectangles must have their corners at grid-points of the original diagram. The corner squares of a subrectangle must be different. The original diagram is a subrectangle of itself.[/i]

I tried to search SRMC problems,but i didn't find them(I found only SRMC 2006).Maybe someone know where on this site i could find SRMC problems?I have all SRMC problems,if someone want i could post them, :wink: Here is one of them,this is one nice inequality from first SRMC: Let $ n$ be an integer with $ n>2$ and $ a_{1},a_{2},\dots,a_{n}\in R^{\plus{}}$.Given any positive integers $ t,k,p$ with $ 1<t<n$,set $ m\equal{}k\plus{}p$,prove the following inequalities: a) $ \frac{a_{1}^{p}}{a_{2}^{k}\plus{}a_{3}^{k}\plus{}\dots\plus{}a_{t}^{k}}\plus{}\frac{a_{2}^{p}}{a_{3}^{k}\plus{}a_{4}^{k}\plus{}\dots\plus{}a_{t\plus{}1}^{k}}\plus{}\dots\plus{}\frac{a_{n\minus{}1}^{p}}{a_{n}^{k}\plus{}a_{1}^{k}\plus{}\dots\plus{}a_{t\minus{}2}^{k}}\plus{}\frac{a_{n}^{p}}{a_{1}^{k}\plus{}a_{2}^{k}\plus{}\dots\plus{}a_{t\minus{}1}^{k}}\geq\frac{(a_{1}^{p}\plus{}a_{2}^{p}\dots\plus{}a_{n}^{p})^{2}}{(t\minus{}1) ( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})}$ b)$ \frac{a_{2}^{k}\plus{}a_{3}^{k}\dots\plus{}a_{t}^{k}}{a_{1}^{p}}\plus{}\frac{a_{3}^{k}\plus{}a_{4}^{k}\dots\plus{}a_{t\plus{}1}^{k}}{a_{2}^{p}}\plus{}\dots\plus{}\frac{a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{t\minus{}1}^{k}}{a_{n}^{p}}\geq\frac{(t\minus{}1)(a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{n}^{k})^{2}}{( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})}$ :wink:

Let $n$ be an integer greater than $1$ and let $X$ be an $n$-element set. A non-empty collection of subsets $A_1, ..., A_k$ of $X$ is tight if the union $A_1 \cup \cdots \cup A_k$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_i$s. Find the largest cardinality of a collection of proper non-empty subsets of $X$, no non-empty subcollection of which is tight. [i]Note[/i]. A subset $A$ of $X$ is proper if $A\neq X$. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.

Let be a continuous function $ f:[0,1]\longrightarrow\left( 0,\infty \right) $ having the property that, for any natural number $ n\ge 2, $ there exist $ n-1 $ real numbers $ 0<t_1<t_2<\cdots <t_{n-1}<1, $ such that $$ \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt. $$ Calculate $ \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} . $

Which one statisfies $n^{29} \equiv 7 \pmod {65}$? $ \textbf{(A)}\ 37 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 55$

For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards. [i]Linus Tang[/i]

Rachel writes down a simple inequality: one $2$-digit number is greater than another. Matt is sitting across from Rachel and peeking at her paper. If Matt, reading upside down, sees a valid inequality between two $2$-digit numbers, compute the number of different inequalities that Rachel could have written. Assume that each digit is either a $1, 6, 8$, or $9$.

If $ x \equal{} \sqrt {1 \plus{} \sqrt {1 \plus{} \sqrt {1 \plus{} \sqrt {1 \plus{} \cdots}}}}$, then: $ \textbf{(A)}\ x \equal{} 1 \qquad\textbf{(B)}\ 0 < x < 1 \qquad\textbf{(C)}\ 1 < x < 2 \qquad\textbf{(D)}\ x\text{ is infinite}$ $ \textbf{(E)}\ x > 2 \text{ but finite}$

Consider the square grid with $A=(0,0)$ and $C=(n,n)$ at its diagonal ends. Paths from $A$ to $C$ are composed of moves one unit to the right or one unit up. Let $C_n$ (n-th catalan number) be the number of paths from $A$ to $C$ which stay on or below the diagonal $AC$. Show that the number of paths from $A$ to $C$ which cross $AC$ from below at most twice is equal to $C_{n+2}-2C_{n+1}+C_n$

Let $$C=\{1,22,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$$ and let $$S=\{4,5,9,14,23,37\}$$ Find two sets $A$ and $B$ with the properties (a) $A\cap B=\emptyset$. (b) $A\cup B=C$. (c) The sum of two distinct elements of $A$ is not in $S$. (d) The sum of two distinct elements of $B$ is not in $S$.

In $2005$ Tycoon Tammy invested $\$100$ for two years. During the the first year her investment suffered a $15\%$ loss, but during the second year the remaining investment showed a $20\%$ gain. Over the two-year period, what was the change in Tammy's investment? $\textbf{(A)}\ 5\%\text{ loss}\qquad \textbf{(B)}\ 2\%\text{ loss}\qquad \textbf{(C)}\ 1\%\text{ gain}\qquad \textbf{(D)}\ 2\% \text{ gain} \qquad \textbf{(E)}\ 5\%\text{ gain}$

Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.

For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class. [i]Proposed by Romania[/i]

Given $ n$ points on the plane, which are the vertices of a convex polygon, $ n > 3$. There exists $ k$ regular triangles with the side equal to $ 1$ and the vertices at the given points. [list][*] Prove that $ k < \frac {2}{3}n$. [*] Construct the configuration with $ k > 0.666n$.[/list]

Given complex numbers $x,y,z$, with $|x|^2+|y|^2+|z|^2=1$. Prove that: $$|x^3+y^3+z^3-3xyz| \le 1$$

A truncated cone has horizontal bases with radii $ 18$ and $ 2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 4\sqrt5 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 6\sqrt3$

Consider the pentagon below. Find its area. [img]https://cdn.artofproblemsolving.com/attachments/7/b/02ad3852b72682513cf62a170ed4aa45c23785.png[/img]

$g$ is a twice differentiable function over the positive reals such that \begin{align}g(x)+2x^3g^\prime(x)+x^4g^{\prime\prime}(x)&= 0 \qquad\text{ for all positive reals } x\\\lim_{x\to\infty}xg(x)&=1\end{align} Find the real number $\alpha>1$ such that $g(\alpha)=1/2$.

Let $n$ be a positive integer. Ana and Banana play a game. Banana thinks of a function $f\colon\mathbb{Z}\to\mathbb{Z}$ and a prime number $p$. He tells Ana that $f$ is nonconstant, $p<100$, and $f(x+p)=f(x)$ for all integers $x$. Ana's goal is to determine the value of $p$. She writes down $n$ integers $x_1,\dots,x_n$. After seeing this list, Banana writes down $f(x_1),\dots,f(x_n)$ in order. Ana wins if she can determine the value of $p$ from this information. Find the smallest value of $n$ for which Ana has a winning strategy. [i]Anthony Wang[/i]

We say that a positive integer is [i]Noble [/i] when: it is composite, it is not divisible by any prime number greater than $20$ and it is not divisible by any perfect cube greater than $1$. How many different Noble numbers are there?