As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$. [asy] size(200); pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C); draw(F--C--A--B--C^^A--D^^B--E); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("84", centroid(H, C, E), fontsize(9.5)); label("35", centroid(H, B, D), fontsize(9.5)); label("30", centroid(H, F, B), fontsize(9.5)); label("40", centroid(H, A, F), fontsize(9.5));[/asy]

Let be four distinct complex numbers $ a,b,c,d $ chosen such that $$ |a|=|b|=|c|=|d|=|b-c|=\frac{|c-d|}{2}=1, $$ and $$ \min_{\lambda\in\mathbb{C}} |a-\lambda d -(1-\lambda )c| =\min_{\lambda\in\mathbb{C}} |b-\lambda d -(1-\lambda )c| . $$ Calculate $ |a-c| $ and $ |a-d|. $ [i]Carmen Botea[/i]

Let $x$ be a positive real number and let $k$ be a positive integer. Assume that $x^k+\frac{1}{x^k}$ and $x^{k+1}+\frac{1}{x^{k+1}}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is also a rational number.

Evaluate \[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\]

[b]p1.[/b] Given an equilateral triangle $ABC$ with area $16\sqrt3$, and an interior point $P$ with distances from vertices $|AP| = 4$ and $|BP| = 6$. (a) Find the length of each side. (b) Find the distance from point $P$ to the side $AB$. (c) Find the distance $|PC|$. [b]p2.[/b] Several players play the following game. They form a circle and each in turn tosses a fair coin. If the coin comes up heads, that player drops out of the game and the circle becomes smaller, if it comes up tails that player remains in the game until his or her next turn to toss. When only one player is left, he or she is the winner. For convenience let us name them $A$ (who tosses first), $B$ (second), $C$ (third, if there is a third), etc. (a) If there are only two players, what is the probability that $A$ (the first) wins? (b) If there are exactly $3$ players, what is the probability that $A$ (the first) wins? (c) If there are exactly $3$ players, what is the probability that $B$ (the second) wins? [b]p3.[/b] A circular castle of radius $r$ is surrounded by a circular moat of width $m$ ($m$ is the shortest distance from each point of the castle wall to its nearest point on shore outside the moat). Life guards are to be placed around the outer edge of the moat, so that at least one life guard can see anyone swimming in the moat. (a) If the radius $r$ is $140$ feet and there are only $3$ life guards available, what is the minimum possible width of moat they can watch? (b) Find the minimum number of life guards needed as a function of $r$ and $m$. [img]https://cdn.artofproblemsolving.com/attachments/a/8/d7ff0e1227f9dcf7e49fe770f7dae928581943.png[/img] [b]p4.[/b] (a)Find all linear (first degree or less) polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all linear polynomials $g(x)$. (b) Prove your answer to part (a). (c) Find all polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all polynomials $g(x)$. (d) Prove your answer to part (c). [b]p5.[/b] A non-empty set $B$ of integers has the following two properties: i. each number $x$ in the set can be written as a sum $x = y+ z$ for some $y$ and $z$ in the set $B$. (Warning: $y$ and $z$ may or may not be distinct for a given $x$.) ii. the number $0$ can not be written as a sum $0 = y + z$ for any $y$ and $z$ in the set $B$. (a) Find such a set $B$ with exactly $6$ elements. (b) Find such a set $B$ with exactly $6$ elements, and such that the sum of all the $6$ elements is $1988$. (c) What is the smallest possible size of such a set $B$ ? (d) Prove your answer to part (c). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Part 1[/u] [b]p1.[/b] Two kids $A$ and $B$ play a game as follows: From a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. What is the sum of all $n$ for which $B$ has a winning strategy? [b]p2.[/b] How many ways can your rearrange the letters of "Alejandro" such that it contains exactly one pair of adjacent vowels? [b]p3.[/b] Assuming real values for $p, q, r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $q + 6i$, and the product of the other two roots is $3 - 4i$. Find the smallest value of $q$. [b]p4.[/b] Lisa has a $3$D box that is $48$ units long, $140$ units high, and $126$ units wide. She shines a laser beam into the box through one of the corners, at a $45^o$ angle with respect to all of the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the eight corners of the box. [u]Part 2[/u] [b]p5.[/b] How many ways can you divide a heptagon into five non-overlapping triangles such that the vertices of the triangles are vertices of the heptagon? [b]p6.[/b] Let $a$ be the greatest root of $y = x^3 + 7x^2 - 14x - 48$. Let $b$ be the number of ways to pick a group of $a$ people out of a collection of $a^2$ people. Find $\frac{b}{2}$ . [b]p7.[/b] Consider the equation $$1 -\frac{1}{d}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},$$ with $a, b, c$, and $d$ being positive integers. What is the largest value for $d$? [b]p8.[/b] The number of non-negative integers $x_1, x_2,..., x_{12}$ such that $$x_1 + x_2 + ... + x_{12} \le 17$$ can be expressed in the form ${a \choose b}$ , where $2b \le a$. Find $a + b$. [u]Part 3[/u] [b]p9.[/b] In the diagram below, $AB$ is tangent to circle $O$. Given that $AC = 15$, $AB = 27/2$, and $BD = 243/34$, compute the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/b/f/b403e5e188916ac4fb1b0ba74adb7f1e50e86a.png[/img] [b]p10.[/b] If $$\left[2^{\log x}\right]^{[x^{\log 2}]^{[2^{\log x}]...}}= 2, $$ where $\log x$ is the base-$10$ logarithm of $x$, then it follows that $x =\sqrt{n}$. Compute $n^2$. [b]p11.[/b] [b]p12.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5, $$ where $n$ is an integer less than $170$. [u]Part 4[/u] [b]p13.[/b] Let $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Define $f(n)$ as the number of distinct two-digit integers that can be formed from digits in $n$. For example, $f(15) = 4$ because the integers $11$, $15$, $51$, $55$ can be formed from digits of $15$. Let $w$ be such that $f(3xz - w) = w$. Find $w$. [b]p14.[/b] Let $w$ be the answer to number $13$ and $z$ be the answer to number $16$. Let $x$ be such that the coefficient of $a^xb^x$ in $(a + b)^{2x}$ is $5z^2 + 2w - 1$. Find $x$. [b]p15.[/b] Let $w$ be the answer to number $13$, $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Let $A$, $B$, $C$, $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. Now, let $AE = 3x$, $ED = w^2 - w + 1$, and $AD = 2z$. If $FG = y$, find $y$. [b]p16.[/b] Let $w$ be the answer to number $13$, and $x$ be the answer to number $16$. Let $z$ be the number of integers $n$ in the set $S = \{w,w + 1, ... ,16x - 1, 16x\}$ such that $n^2 + n^3$ is a perfect square. Find $z$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that there are $1977$ non-similar triangles such that the angles $A, B, C$ satisfy $\frac{\sin A + \sin B + \sin C}{\cos A +\cos B + \cos C} = \frac{12}{7}$ and $\sin A \sin B \sin C = \frac{12}{25}$.

Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$

In a group $G$, we have two elements $x,y$ such that $x^{n}=e,y^2=e,yxy=x^{-1}$, $n\ge 1$. Prove that for any $k\in\mathbb{N}$ holds $(x^ky)^2=e$. Note : e=group's identity .

A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.

Let $x, y$ and $z$ be rational numbers satisfying $$x^3 + 3y^3 + 9z^3 - 9xyz = 0.$$ Prove that $x = y = z = 0$.

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.

Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(1) \neq 1$. For a positive integer $n$, define $\text{divs}(n)$ to be the set of positive divisors of $n$. A positive integer $m$ is $f$-cool if there exists a positive integer $n$ for which $$f[\text{divs}(m)]=\text{divs}(n).$$ Prove that for any such $f$, there are finitely many $f$-cool integers. (The notation $f[S]$ for some set $S$ denotes the set $\{f(s):s \in S\}$.)

Let $a, b, c$ be the side lengths of an non-degenerate triangle with $a \le b \le c$. With $t (a, b, c)$ denote the minimum of the quotients $\frac{b}{a}$ and $\frac{c}{b}$ . Find all values that $t (a, b, c)$ can take.

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the inequality \[ f(y) - \left(\frac{z-y}{z-x} f(x) + \frac{y-x}{z-x}f(z)\right) \leq f\left(\frac{x+z}{2}\right) - \frac{f(x)+f(z)}{2} \] for all real numbers $x < y < z$. [i]Proposed by Gabriel Carroll[/i]

The positive integer $8833$ has the property that $8833 = 88^2 + 33^2.$ Find the (unique) other four-digit positive integer $\overline{abcd}$ where $\overline{abcd} = (\overline{ab})^2 + (\overline{cd})^2.$ [i]Proposed by Allen Yang[/i]

Find all real solutions of the system of equations: \[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$

Let $ \alpha\equal{}2\plus{}\sqrt{3}$. Prove that $ \alpha^n\minus{}[\alpha^n]\equal{}1\minus{}\alpha^{\minus{}n}$ for all $ n \in \mathbb{N}_0$.

Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$: $a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$ if $a_0=2020$, determine the value of $a_1$.

We call the sequence $d_1,....,d_n$ of natural numbers, not necessarily distinct, [b]covering[/b] if there exists arithmetic progressions like $c_1+kd_1$,....,$c_n+kd_n$ such that every natural number has come in at least one of them. We call this sequence [b]short[/b] if we can not delete any of the $d_1,....,d_n$ such that the resulting sequence be still covering. [b]a)[/b] Suppose that $d_1,....,d_n$ is a short covering sequence and suppose that we've covered all the natural numbers with arithmetic progressions $a_1+kd_1,.....,a_n+kd_n$, and suppose that $p$ is a prime number that $p$ divides $d_1,....,d_k$ but it does not divide $d_{k+1},....,d_n$. Prove that the remainders of $a_1,....,a_k$ modulo $p$ contains all the numbers $0,1,.....,p-1$. [b]b)[/b] Write anything you can about covering sequences and short covering sequences in the case that each of $d_1,....,d_n$ has only one prime divisor. [i]proposed by Ali Khezeli[/i]

The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for $\forall$ $x\in (-\infty , m]$.

Find all triples $ (x,y,z)$ of positive real numbers which satisfy $ 2x^3 \equal{} 2y(x^2 \plus{} 1) \minus{} (z^2 \plus{} 1)$; $ 2y^4 \equal{} 3z(y^2 \plus{} 1) \minus{} 2(x^2 \plus{} 1)$; $ 2z^5 \equal{} 4x(z^2 \plus{} 1) \minus{} 3(y^2 \plus{} 1)$.

Let $f: \mathbb{R}^{2} \to \mathbb{R}^{+}$such that for every rectangle $A B C D$ one has $$ f(A)+f(C)=f(B)+f(D). $$ Let $K L M N$ be a quadrangle in the plane such that $f(K)+f(M)=f(L)+f(N)$, for each such function. Prove that $K L M N$ is a rectangle. [i]Proposed by Navid.[/i]

Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold [list] [*] $a+b \in F$, and [*] $\gcd(P(a),P(b))=1$. [/list] Prove that $P(x)$ is a constant polynomial.

Find all possible values of $\frac{1}{x}+\frac{1}{y}$, if $x,y$ are real numbers not equal to $0$ that satisfy $$x^3+y^3+3x^2y^2=x^3y^3$$