Prove that exists triangle that can be partitions in $2050$ congruent triangles.

Let n be a positive integer, and let $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ be positive real numbers such that $a_1\geq a_2\geq ...\geq a_n$ and $b_1\geq a_1$, $b_1b_2\geq a_1a_2$, $b_1b_2b_3\geq a_1a_2a_3$, ..., $b_1b_2...b_n\geq a_1a_2...a_n$. Prove that $b_1+b_2+...+b_n\geq a_1+a_2+...+a_n$.

Two circles $C_{1}$ and $C_{2}$ intersect in $A$ and $B$. A line passing through $B$ intersects $C_{1}$ in $C$ and $C_{2}$ in $D$. Another line passing through $B$ intersects $C_{1}$ in $E$ and $C_{2}$ in $F$, $(CF)$ intersects $C_{1}$ and $C_{2}$ in $P$ and $Q$ respectively. Make sure that in your diagram, $B, E, C, A, P \in C_{1}$ and $B, D, F, A, Q \in C_{2}$ in this order. Let $M$ and $N$ be the middles of the arcs $BP$ and $BQ$ respectively. Prove that if $CD=EF$, then the points $C,F,M,N$ are cocylic in this order.

Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron. [i]Stere Ianuș[/i]

An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.

Let $A$ be a set of functions $f : R\to R$. For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$. Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.

A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.

Let $ABC$ be a nonequilateral triangle. Prove that there exist two points $P$ and $Q$ in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of $ABC$, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.

Define Fibonacci sequence $\{F\}_{n=0}^{\infty}$ as $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n +F_{n-1}$ for every integer $n > 1$. Determine all quadruples $(a, b, c,n)$ of positive integers with a $< b < c$ such that each of $a, b,c,a + n, b + n,c + 2n$ is a term of the Fibonacci sequence.

Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$ and the graphs of the functions $y =\sqrt{x^3+ 1}$ and $y = - \sqrt[3]{x^2+ 2x}$.

a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying: [b]i.[/b] each has $k$ elements; [b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$) [b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements. b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.

The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$? $ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$

A circle of radius $a$ rolls in the plane along the $x$-axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter $a$, likewise rolling in the plane along the $x$-axis.

Find the number of three-digit positive integers which have three distinct digits where the sum of the digits is an even number such as 925 and 824.

Let $n$ be the product of the first $10$ primes, and let $$S=\sum_{xy\mid n} \varphi(x) \cdot y,$$ where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $xy$ divides $n$. Compute $\tfrac{S}{n}.$

Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression: \[p(1)+p(2)+p(3)+...+p(999).\]

Let $ABC$ be a triangle with incenter $I$. The line through $I$, perpendicular to $AI$, intersects the circumcircle of $ABC$ at points $P$ and $Q$. It turns out there exists a point $T$ on the side $BC$ such that $AB + BT = AC + CT$ and $AT^2 = AB \cdot AC$. Determine all possible values of the ratio $IP/IQ$.

Prove:If $ a\equal{}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{n}^{\alpha_n}$ is a perfect number, then $ 2<\prod_{i\equal{}1}^n\frac{p_i}{p_i\minus{}1}<4$ ; if moreover, $ a$ is odd, then the upper bound $ 4$ may be reduced to $ 2\sqrt[3]{2}$.

7. In triangle $ABC$, let $N$ and $M$ be the midpoints of $AB$ and $AC$, respectively. Point $P$ is chosen on the arc $BC$ not containing $A$ of the circumcircle of $ABC$ such that $BNMP$ is cyclic. Given $BC=28$, $AC=30$ and $AB = 26$, the value of $AP$ may be expressed as $m/\sqrt{n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Compute $m+n$. [i]Proposed by winnertakeover[/i]

Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$. If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?

for any positive integer $n$ greater than $1$, show that \[2^n<\binom{2n}{n}<\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}\]

Given $X=\{0,a,b,c\}$, let $M(X)=\{f|f: X\to X\}$ denote the set of all functions from $X$ into itself. An addition table on $X$ is given us follows: $+$ $0$ $a$ $b$ $c$ $0$ $0$ $a$ $b$ $c$ $a$ $a$ $0$ $c$ $b$ $b$ $b$ $c$ $0$ $a$ $c$ $c$ $b$ $a$ $0$ a)If $S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}$, find $|S|$. b)If $I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}$, find $|I|$.

We have a deck of $2n$ cards. Each shuffling changes the order from $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ to $a_1,b_1,a_2,b_2,\ldots,a_n,b_n$. Determine all even numbers $2n$ such that after shuffling the deck $8$ times the original order is restored.

Heesu plays a game where he starts with $1$ piece of candy. Every turn, he flips a fair coin. On heads, he gains another piece of candy, unless he already has $5$ pieces of candy, in which case he loses $4$ pieces of candy and goes back to having $1$ piece of candy. On tails, the game ends. What is the expected number of pieces of candy that Heesu will have when the game ends?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].