Prove that for every partition of $ \{ 1,2,3,4,5,6,7,8,9\} $ into two subsets, one of the subsets contains three numbers such that the sum of two of them is equal to the double of the third.

Calculate $1 + 3 + 5 +\cdots+ 195 + 197 + 199$

Sequence of positive integers $\{x_k\}_{k\geq 1}$ is given such that $x_1=1$ and for all $n\geq 1$ we have $$x_{n+1}^2+P(n)=x_n x_{n+2}$$ where $P(x)$ is a polynomial with non-negative integer coefficients. Prove that $P(x)$ is the constant polynomial. Proposed by [i]Navid Safaei[/i]

Solve the equation $[x] \cdot \{x\} = 1991x$. (Here $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$.)

A scout troop buys $ 1000$ candy bars at a price of five for $ \$2$. They sell all the candy bars at a price of two for $ \$1$. What was their profit, in dollars? $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 200 \qquad \textbf{(C)}\ 300 \qquad \textbf{(D)}\ 400 \qquad \textbf{(E)}\ 500$

Lautaro, Camilo and Rafael give the same exams. Each note is a positive integer. Camilo was the first in physics. Lautaro obtained a total score of $20$, Camilo, a total of $10$ and Rafael, a total of $9$. Among all the tests, there were no two scores that were repeated. Determine how many They took exams, and who was second in math.

Let $ \sigma = (a_1, a_2, \cdots , a_n)$ be permutation of $ (1, 2 ,\cdots, n)$. A pair $ (a_i, a_j)$ is said to correspond to an [b]inversion[/b] of $\sigma$ if $ i<j$ but $ a_i>a_j$. How many permutations of $ (1,2,\cdots,n)$, $ n \ge 3$, have exactly [b]two[/b] inversions? For example, In the permutation $(2,4,5,3,1)$, there are 6 inversions corresponding to the pairs $ (2,1),(4,3),(4,1),(5,3),(5,1),(3,1)$.

For any positive real numbers $a, b, c, d$, prove the following inequality: $$(a^2+b^2)(b^2+c^2)(c^2+d^2)(d^2+a^2) \geq 64abcd|(a-b)(b-c)(c-d)(d-a)|$$ [i]Proposed by Anton Trygub[/i]

Let $(a_1, a_2,..., a_n)$ and $(b_1, b_2, ..., b_n)$ be two sequences of positive real numbers. Let $\pi$ be a permutation of the set $\{1, 2, ..., n\}$, for which the sum $a_{\pi(1)}(b_{\pi(1)}+b_{\pi(2)}+...+b_{\pi(n)})+a_{\pi(2)}(b_{\pi(3)}+b_{\pi(3)}+...+b_{\pi(n)})+...+a_{\pi(n)}b_{\pi(n)}$ is minimal. Proce for this permutation $\pi$, that $$ \frac{a_{\pi(1)}}{b_{\pi(1)}}\le \frac{a_{\pi(2})}{b_{\pi(2)}}\le ... \le \frac{a_{\pi(n)}}{b_{\pi(n)}}$$ Application: In an idealized role-playing game you fight against $n$ opponents at the same time. In order to minimize the damage you suffer yourself, you should first take care of your opponent for the ratio of the time it takes to defeat him (if you only focus on him), and the damage it does per second is minimal; next, one should fight the opponent with the second smallest such ratio, and so on.

Consider the sequence $a(n)$ defined by the following conditions:$$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$ How many perfect squares no greater in value than $1000 000$ will be found among the first terms of the sequence? ( (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.) (A Andjans)

Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$

Two players take turns entering a symbol in an empty cell of a $1 \times n$ chessboard, where $n$ is an integer greater than $1$. Aaron always enters the symbol $X$ and Betty always enters the symbol $O$. Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes first, which player has a winning strategy?

Write two ones, then a $2$ between them, then a $3$ between the numbers whose sum is $3$, then a $4$ between the numbers whose sum is $4$, as shown below: $$(1, 1), (1, 2, 1),(1, 3, 2, 3, 1), (1, 4, 3, 2, 3, 4, 1)$$ and so on. Prove that the number of times $n$ appears, ($n\ge 2$), is equal to the number of positive integers less than $n$ and relative prime with $n$..

Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.

Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

For any positive integer $d$, prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number.

Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$. Find $f(1987)$. [i]Proposed by Australia.[/i]

Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16,$ $AB=107,$ $FG=17,$ and $EF=184,$ what is the length of $CE$? [asy] unitsize(1 cm); pair A, B, C, D, E, F, G, H; A = (0,0); B = (5,0); C = (5,1.5); D = (0,1.5); E = (1,1.5); F = (8,1.5); G = (8,3.5); H = (1,3.5); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); dot("A", A, SW); dot("B", B, SE); dot("C", C, SE); dot("D", D, NW); dot("E", E, NW); dot("F", F, SE); dot("G", G, NE); dot("H", H, NW); [/asy]

Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions: (1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$; (2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$; (2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$. Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.

find all positive integer $a\geq 2 $ and $b\geq2$ such that $a$ is even and all the digits of $a^b+1$ are equals.

In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$

If $p,q,r$ are the roots of the cubic equation $x^3 - 3px^2 + 3q^2 x - r^3 = 0$, then show that $p = q =r$.

Given distinct straight lines $ OA$ and $ OB$. From a point in $ OA$ a perpendicular is drawn to $ OB$; from the foot of this perpendicular a line is drawn perpendicular to $ OA$. From the foot of this second perpendicular a line is drawn perpendicular to $ OB$; and so on indefinitely. The lengths of the first and second perpendiculars are $ a$ and $ b$, respectively. Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is: $ \textbf{(A)}\ \frac {b}{a \minus{} b} \qquad \textbf{(B)}\ \frac {a}{a \minus{} b} \qquad \textbf{(C)}\ \frac {ab}{a \minus{} b} \qquad \textbf{(D)}\ \frac {b^2}{a \minus{} b} \qquad \textbf{(E)}\ \frac {a^2}{a \minus{} b}$

Find all functions $f:\mathbb R\rightarrow \mathbb R$ such that for all real numbers $x,y$: $f(x)f(y)+f(xy)\leq x+y$.

Let $p$ be a prime number such that $p\mid (2^{2019}-1) .$ The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}$ $(n\geq 1).$ Prove that $p\nmid (a_n+1),$ for any $n\geq 0.$