Found problems: 85335
[asy]
real s=sqrt(3)/2;
draw(box((0,0),(1,1)));
draw((1+s,0.5)--(1,1));
draw((1+s,0.5)--(1,0));
draw((0,1)--(1+s,0.5));
label("$A$",(1,1),N);
label("$B$",(1,0),S);
label("$C$",(0,0),W);
label("$D$",(0,1),W);
label("$E$",(1+s,0.5),E);
//Credit to TheMaskedMagician for the diagram
[/asy]
In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees?
$\textbf{(A) }10\qquad\textbf{(B) }12.5\qquad\textbf{(C) }15\qquad\textbf{(D) }20\qquad\textbf{(E) }25$
Let $S=\{1,2,\ldots,10\}$. Three numbers are chosen with replacement from $S$. If the chosen numbers denote the lengths of sides of a triangle, then the probability that they will form a triangle is:
$\textbf{(A)}~\frac{101}{200}$
$\textbf{(B)}~\frac{99}{200}$
$\textbf{(C)}~\frac12$
$\textbf{(D)}~\frac{110}{200}$
We are given an acute triangle $ABC$, and inside it a point $P$, which is not on any of the heights $AA_1$, $BB_1$, $CC_1$. The rays $AP$, $BP$, $CP$ intersect the circumcircle of $ABC$ at points $A_2$, $B_2$, $C_2$. Prove that the circles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$ are concurrent.
$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds:
$\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$
Let $n$ be a positive integer and let $x_1, x_2, \ldots, x_n$ be positive real numbers such that $x_1x_2 \cdots x_n = 1$. Prove that \[\displaystyle\sum_{i=1}^n x_i^n (1 + x_i) \geq \dfrac{n}{2^{n-1}} \prod_{i=1}^n (1 + x_i).\]
[i]IMO Shortlist[/i]
$2023$ wise men are located in a circle. Each of them thinks either that the earth is the center of the universe, or that it is not. Once a minute, all the wise men express their opinion at the same time. Every wise man who is between two wise men with an opinion different from his will change his mind at that moment. The others don't change their minds. The others don't change their minds. Determine the smallest necessary time for all the wise men to have the same opinion, without regardless of initial opinions or your location.
Prove that the sums of the opposite dihedral angles of a tetrahedron are equal if and only if the sums of the opposite edges of this tetrahedron are equal.
The complex numbers $z$ and $w$ satisfy the system
\begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*}
Find the smallest possible value of $|zw|^2$.
Find $k$ where $2^k$ is the largest power of $2$ that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\]
The sequence $a_1, a_2,...$ is de fined by the equalities $a_1 = 2, a_2 = 12$ and $a_{n+1} = 6a_n-a_{n-1}$ for every positive integer $n \ge 2$. Prove that no member of this sequence is equal to a perfect power (greater than one) of a positive integer.
Isosceles triangle $ABC$ has angle $\angle BAC = 135^o$ and $AB = 2$. What is its area?
Prove the following inequality.
\[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]
A basket is called "[i]Stuff Basket[/i]" if it includes $10$ kilograms of rice and $30$ number of eggs. A market is to distribute $100$ Stuff Baskets. We know that there is totally $1000$ kilograms of rice and $3000$ number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide $100$ Stuff Baskets?
Let $f:[0,1]\to\mathbb R$ be a given continuous function. Find the limit
$$\lim_{n\to\infty}(n+1)\sum_{k=0}^n\int^1_0x^k(1-x)^{n-k}f(x)dx.$$
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
[i]Proposed by Lewis Chen[/i]
Three circles in the plane, whose interiors have no common point, meet each other at three pairs of points: $A_1$ and $A_2$, $B_1$ and $B_2$, and $C_1$ and $C_2$, where points $A_2,B_2,C_2$ lie inside the triangle $A_1B_1C_1$. Prove that
\[A_1B_2 \cdot B_1C_2 \cdot C_1A_2 = A_1C_2 \cdot C_1B_2 \cdot B_1A_2 .\]
$n_{1},\ldots ,n_{k}, a$ are integers that satisfies the above conditions A)For every $i\neq j$, $(n_{i}, n_{j})=1$ B)For every $i, a^{n_{i}}\equiv 1 (mod n_{i})$ C)For every $i, X^{a-1}\equiv 0(mod n_{i})$.
Prove that $a^{x}\equiv 1(mod x)$ congruence has at least $2^{k+1}-2$ solutions. ($x>1$)
Each vertex on a cube is colored black or white independently at random with equal probability. What is the expected number of edges on the cube that connect vertices of different colors?
(A.Zaslavsky, 9--10) Quadrilateral $ ABCD$ is circumscribed arounda circle with center $ I$. Prove that the projections of points $ B$ and $ D$ to the lines $ IA$ and $ IC$ lie on a single circle.
The simplest form of $ 1 \minus{} \frac{1}{1 \plus{} \frac{a}{1 \minus{} a}}$ is:
$ \textbf{(A)}\ {a}\text{ if }{a\not\equal{} 0} \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ {a}\text{ if }{a\not\equal{} \minus{}1}\qquad
\textbf{(D)}\ {1 \minus{} a}\text{ with not restriction on }{a}\qquad
\textbf{(E)}\ {a}\text{ if }{a\not\equal{} 1}$
Define the [i]factorial function[/i] of $n,$ denoted $\partial (n),$ as the sum of the factorials of the digits of $n.$ For example, $\partial(2024)=2!+0!+2!+4!=29.$ There are four positive integers such that $\partial(\partial(n))=n$ and $\partial(n) \neq n.$ Given that $n=871$ is one of them, compute the sum of the other three.
A cube of side length $n$ is made up of $n^3$ smaller unit cubes. Some of the six faces of the large cube are fully painted. When the large cube is taken apart, 245 smaller cubes do not have any paint on them.
Determine the value(s) of $n$ and how many faces of the large cube were painted.
On a straight line lie $100$ points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these $101$ points?
Let $ABC$ an acutangle triangle with orthocenter $H$ and circumcenter $O$. Let $D$ the intersection of $AO$ and $BH$. Let $P$ be the point on $AB$ such that $PH=PD$. Prove that the points $B, D, O$ and $P$ lie on a circle.
a) Prove that a convex $n$-gon cannot have more than $3$ interior angles acute.
b) Prove that a convex $n$-gon that has $3$ interior angles equal to $60^0,$ is equilateral.