Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as: $u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$ a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$

Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that: (a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.

Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.

[u]Round 5[/u] [b]p13.[/b] Given that $x^3 + y^3 = 208$ and $x + y = 4$, what is the value of $\frac{1}{x} +\frac{1}{y}$? [b]p14.[/b] Find the sum of all three-digit integers $n$ such that the value of $n$ is equal to the sum of the factorials of $n$’s digits. [b]p15.[/b] Three christmas lights are initially off. The Grinch decides to fiddle around with the lights, switching one of the lights each second. He wishes to get every possible combination of lights. After how many seconds can the Grinch complete his task? [u]Round 6[/u] [b]p16.[/b] A regular tetrahedron of side length $1$ has four similar tetrahedrons of side length $1/2$ chopped off, one from each of the four vertices. What is the sum of the numbers of vertices, edges, and faces of the remaining solid? [b]p17.[/b] Mario serves a pie in the shape of a regular $2013$-gon. To make each slice, he must cut in a straight line starting from one vertex and ending at another vertex of the pie. Every vertex of a slice must be a vertex of the original $2013$-gon. If every person eats at least one slice of pie regardless of the size, what is the maximum number of people the $2013$-gon pie can feed? [b]p18.[/b] Find the largest integer $x$ such that $x^2 + 1$ divides $x^3 + x - 1000$. [u]Round 7[/u] [b]p19.[/b] In $\vartriangle ABC$, $\angle B = 87^o$, $\angle C = 29^o$, and $AC = 37$. The perpendicular bisector of $\overline{BC}$ meets $\overline{AC}$ at point $T$. What is the value of $AB + BT$? [b]p20.[/b] Consider the sequence $f(1) = 1$, $f(2) = \frac12$ ,$ f(3) =\frac{1+3}{2}$, $f(4) =\frac{ 1+3}{2+4}$ ,$ f(5) = \frac{ 1+3+5}{2+4} . . . $ What is the minimum value of $n$, with $n > 1$, such that $|f(n) - 1| \le \frac{1}{10 }$. [b]p21.[/b] Three unit circles are centered at $(0, 0)$,$(0, 2)$, and $(2, 0)$. A line is drawn passing through $(0, 1)$ such that the region inside the circles and above the line has the same area as the region inside the circles and below the line. What is the equation of this line in $y = mx + b$ form? [u]Round 8[/u] [b]p22.[/b] The two walls of a pinball machine are positioned at a $45$ degree angle to each other. A pinball, represented by a point, is fired at a wall (but not at the intersection of the two walls). What is the maximum number of times the ball can bounce off the walls? [b]p23.[/b] Albert is fooling people with his weighted coin at a carnival. He asks his guests to guess how many times heads will show up if he flips the coin $4$ times. Richard decides to play the game and guesses that heads will show up $2$ times. In the previous game, Zach guessed that the heads would show up 3 times. In Zach’s game, there were least 3 heads, and given this information, Zach had a $\frac49$ chance of winning. What is the probability that Richard guessescorrectly? [b]p24.[/b] Let $S$ be the set of all positive integers relatively prime to $2013$ that have no prime factor greater than $15$. Find the sum of the reciprocals of all of the elements of $S$. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134546p28406927]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $a,b,c,x$ are real numbers such that $abc \not= 0$ and \[ \frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c}, \] then prove that $a = b = c$.

Let $n>1$ be a natural number. Find the real values of the parameter $a$, for which the equation $\sqrt[n]{1+x}+\sqrt[n]{1-x}=a$ has a single real root.

We are given coins of $1,2,5,10,20,50$ lipas and of $1$ kuna (Croatian currency: $1$ kuna = $100$ lipas). Prove that if a bill of $M$ lipas can be paid by $N$ coins, then a bill of $N$ kunas can be paid by M coins.

[b]p1.[/b] Find $20 \cdot 18 + 20 + 18 + 1$. [b]p2.[/b] Suzie’s Ice Cream has $10$ flavors of ice cream, $5$ types of cones, and $5$ toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s? [b]p3.[/b] Let $a = 7$ and $b = 77$. Find $\frac{(2ab)^2}{(a+b)^2-(a-b)^2}$ . [b]p4.[/b] Sebastian invests $100,000$ dollars. On the first day, the value of his investment falls by $20$ percent. On the second day, it increases by $25$ percent. On the third day, it falls by $25$ percent. On the fourth day, it increases by $60$ percent. How many dollars is his investment worth by the end of the fourth day? [b]p5.[/b] Square $ABCD$ has side length $5$. Points $K,L,M,N$ are on segments $AB$,$BC$,$CD$,$DA$ respectively,such that $MC = CL = 2$ and $NA = AK = 1$. The area of trapezoid $KLMN$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Suppose that $p$ and $q$ are prime numbers. If $p + q = 30$, find the sum of all possible values of $pq$. [b]p7.[/b] Tori receives a $15 - 20 - 25$ right triangle. She cuts the triangle into two pieces along the altitude to the side of length $25$. What is the difference between the areas of the two pieces? [b]p8.[/b] The factorial of a positive integer $n$, denoted $n!$, is the product of all the positive integers less than or equal to $n$. For example, $1! = 1$ and $5! = 120$. Let $m!$ and $n!$ be the smallest and largest factorial ending in exactly $3$ zeroes, respectively. Find $m + n$. [b]p9.[/b] Sam is late to class, which is located at point $B$. He begins his walk at point $A$ and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance? [img]https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png[/img] [b]p10.[/b] Mr. Iyer owns a set of $6$ antique marbles, where $1$ is red, $2$ are yellow, and $3$ are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining $4$ out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positiveintegers. What is $m + n$? [b]p11.[/b] If $a$ is a positive integer, what is the largest integer that will always be a factor of $(a^3+1)(a^3+2)(a^3+3)$? [b]p12.[/b] What is the largest prime number that is a factor of $160,401$? [b]p13.[/b] For how many integers $m$ does the equation $x^2 + mx + 2018 = 0$ have no real solutions in $x$? [b]p14.[/b] What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is $7887887$. [b]p15.[/b] In circle $\omega$ inscribe quadrilateral $ADBC$ such that $AB \perp CD$. Let $E$ be the intersection of diagonals $AB$ and $CD$, and suppose that $EC = 3$, $ED = 4$, and $EB = 2$. If the radius of $\omega$ is $r$, then $r^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $m + n$. [b]p16.[/b] Suppose that $a, b, c$ are nonzero real numbers such that $2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca$. Find the value of $\frac{9(a + b + c)^3}{5abc}$ . [b]p17.[/b] Call a positive integer n spicy if there exist n distinct integers $k_1, k_2, ... , k_n$ such that the following two conditions hold: $\bullet$ $|k_1| + |k_2| +... + |k_n| = n2$, $\bullet$ $k_1 + k_2 + ...+ k_n = 0$. Determine the number of spicy integers less than $10^6$. [b]p18.[/b] Consider the system of equations $$|x^2 - y^2 - 4x + 4y| = 4$$ $$|x^2 + y^2 - 4x - 4y| = 4.$$ Find the sum of all $x$ and $y$ that satisfy the system. [b]p19.[/b] Determine the number of $8$ letter sequences, consisting only of the letters $W,Q,N$, in which none of the sequences $WW$, $QQQ$, or $NNNN$ appear. For example, $WQQNNNQQ$ is a valid sequence, while $WWWQNQNQ$ is not. [b]p20.[/b] Triangle $\vartriangle ABC$ has $AB = 7$, $CA = 8$, and $BC = 9$. Let the reflections of $A,B,C$ over the orthocenter H be $A'$,$B'$,$C'$. The area of the intersection of triangles $ABC$ and $A'B'C'$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ , where $b$ is squarefree and $a$ and $c$ are relatively prime. determine $a+b+c$. (The orthocenter of a triangle is the intersection of its three altitudes.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are prime numbers $a$, $b$, and $c$ such that the system of equations $$a \cdot x - 3 \cdot y + 6 \cdot z = 8$$ $$b \cdot x + 3\frac12 \cdot y + 2\frac13 \cdot z = -28$$ $$c \cdot x - 5\frac12 \cdot y + 18\frac13 \cdot z = 0$$ has infinitely many solutions for $(x, y, z)$. Find the product $a \cdot b \cdot c$.

Consider the matrices $A\in \mathcal{M}_{m,n}(\mathbb{C})$ and $B\in \mathcal{M}_{n,m}(\mathbb{C})$ with $n\le m$. It is given that $\text{rank}(AB)=n$ and $(AB)^2=AB$. a)Prove that $(BA)^3=(BA)^2$. b)Find $BA$.

Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that \[ \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x \] for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

Find all polynomials $f(x)$ such that $f(2x) = f'(x) f''(x)$.

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a^3}{a^2 + bc}+\frac{b^3}{b^2 + ca}+\frac{c^3}{c^2 + ab} \ge \frac{(a^2 + b^2 + c^2)(ab + bc + ca)}{a^3 + b^3 + c^3 + 3abc}$$

Prove that there are two numbers $u$ and $v$, between any $101$ real numbers that apply $100 |u - v| \cdot |1 - uv| \le (1 + u^2)(1 + v^2)$

Determine all functions $f:\mathbb R\rightarrow\mathbb R$ satisfying the condition \[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\] for all real numbers $x$ and $y$.

For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \le 20$ and $r_1 + r_2 = \sqrt{2}$.

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

Answer the following questions : $\textbf{(a)}$ Find all real solutions of the equation $$\Big(x^2-2x\Big)^{x^2+x-6}=1$$ Explain why your solutions are the only solutions. $\textbf{(b)}$ The following expression is a rational number. Find its value. $$\sqrt[3]{6\sqrt{3}+10} -\sqrt[3]{6\sqrt{3}-10}$$

Find the greatest integer which doesn't exceed $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$ (A)$81$ (B)$80$ (C)$79$ (D)$82$

Prove that for every integer $n$ ($n > 1$) there exist two positive integers $x$ and $y$ ($x \leq y$) such that $$\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}$$

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

We denote $N_{2010}=\{1,2,\cdots,2010\}$ [b](a)[/b]How many non empty subsets does this set have? [b](b)[/b]For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products? [b](c)[/b]Same question as the [b](b)[/b] part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$. Albanian National Mathematical Olympiad 2010---12 GRADE Question 2.

We say that a strictly increasing sequence of positive integers $n_1, n_2,\ldots$ is [i]non-decelerating[/i] if $n_{k+1}-n_k\le n_{k+2}-n_{k+1}$ holds for all positive integers $k$. We say that a strictly increasing sequence $n_1, n_2, \ldots$ is [i]convergence-inducing[/i], if the following statement is true for all real sequences $a_1, a_2, \ldots$: if subsequence $a_{m+n_1}, a_{m+n_2}, \ldots$ is convergent and tends to $0$ for all positive integers $m$, then sequence $a_1, a_2, \ldots$ is also convergent and tends to $0$. Prove that a non-decelerating sequence $n_1, n_2,\ldots$ is convergence-inducing if and only if sequence $n_2-n_1$, $n_3-n_2$, $\ldots$ is bounded from above. [i]Proposed by András Imolay[/i]

Let $a$ and  $b$ be the roots of the equation $x^2 + x - 3 = 0$. Find the value of the expression $4b^2 -a^3$.

Let $a$ and $b$ be real numbers, where $a \not= 0$ and $a \not= b$ and all the roots of the equation $ax^{3}-x^{2}+bx-1 = 0$ is a real and positive number. Find the smallest possible value of $P = \dfrac{5a^{2}-3ab+2}{a^{2}(b-a)}$. [i](Heir of Ramanujan)[/i]