Find the sum of the smallest 20 positive real solutions of the equation $\sin x=\cos 2x .$

Let $a, b, c$ be positive real numbers such that: $$ab - c = 3$$ $$abc = 18$$ Calculate the numerical value of $\frac{ab}{c}$

[b]p1.[/b] What is the units digit of $1 + 9 + 9^2 +... + 9^{2015}$ ? [b]p2.[/b] In Fourtown, every person must have a car and therefore a license plate. Every license plate must be a $4$-digit number where each digit is a value between $0$ and $9$ inclusive. However $0000$ is not a valid license plate. What is the minimum population of Fourtown to guarantee that at least two people who have the same license plate? [b]p3.[/b] Two sides of an isosceles triangle $\vartriangle ABC$ have lengths $9$ and $4$. What is the area of $\vartriangle ABC$? [b]p4.[/b] Let $x$ be a real number such that $10^{\frac{1}{x}} = x$. Find $(x^3)^{2x}$. [b]p5.[/b] A Berkeley student and a Stanford student are going to visit each others campus and go back to their own campuses immediately after they arrive by riding bikes. Each of them rides at a constant speed. They first meet at a place $17.5$ miles away from Berkeley, and secondly $10$ miles away from Stanford. How far is Berkeley away from Stanford in miles? [b]p6.[/b] Let $ABCDEF$ be a regular hexagon. Find the number of subsets $S$ of $\{A,B,C,D,E, F\}$ such that every edge of the hexagon has at least one of its endpoints in $S$. [b]p7.[/b] A three digit number is a multiple of $35$ and the sum of its digits is $15$. Find this number. [b]p8.[/b] Thomas, Olga, Ken, and Edward are playing the card game SAND. Each draws a card from a $52$ card deck. What is the probability that each player gets a dierent rank and a different suit from the others? [b]p9.[/b] An isosceles triangle has two vertices at $(1, 4)$ and $(3, 6)$. Find the $x$-coordinate of the third vertex assuming it lies on the $x$-axis. [b]p10.[/b] Find the number of functions from the set $\{1, 2,..., 8\}$ to itself such that $f(f(x)) = x$ for all $1 \le x \le 8$. [b]p11.[/b] The circle has the property that, no matter how it's rotated, the distance between the highest and the lowest point is constant. However, surprisingly, the circle is not the only shape with that property. A Reuleaux Triangle, which also has this constant diameter property, is constructed as follows. First, start with an equilateral triangle. Then, between every pair of vertices of the triangle, draw a circular arc whose center is the $3$rd vertex of the triangle. Find the ratio between the areas of a Reuleaux Triangle and of a circle whose diameters are equal. [b]p12.[/b] Let $a$, $b$, $c$ be positive integers such that gcd $(a, b) = 2$, gcd $(b, c) = 3$, lcm $(a, c) = 42$, and lcm $(a, b) = 30$. Find $abc$. [b]p13.[/b] A point $P$ is inside the square $ABCD$. If $PA = 5$, $PB = 1$, $PD = 7$, then what is $PC$? [b]p14.[/b] Find all positive integers $n$ such that, for every positive integer $x$ relatively prime to $n$, we have that $n$ divides $x^2 - 1$. You may assume that if $n = 2^km$, where $m$ is odd, then $n$ has this property if and only if both $2^k$ and $m$ do. [b]p15.[/b] Given integers $a, b, c$ satisfying $$abc + a + c = 12$$ $$bc + ac = 8$$ $$b - ac = -2,$$ what is the value of $a$? [b]p16.[/b] Two sides of a triangle have lengths $20$ and $30$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side? [b]p17.[/b] Find the number of non-negative integer solutions $(x, y, z)$ of the equation $$xyz + xy + yz + zx + x + y + z = 2014.$$ [b]p18.[/b] Assume that $A$, $B$, $C$, $D$, $E$, $F$ are equally spaced on a circle of radius $1$, as in the figure below. Find the area of the kite bounded by the lines $EA$, $AC$, $FC$, $BE$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/57e6e1c4ef17f84a7a66a65e2aa2ab9c7fd05d.png[/img] [b]p19.[/b] A positive integer is called cyclic if it is not divisible by the square of any prime, and whenever $p < q$ are primes that divide it, $q$ does not leave a remainder of $1$ when divided by $p$. Compute the number of cyclic numbers less than or equal to $100$. [b]p20.[/b] On an $8\times 8$ chess board, a queen can move horizontally, vertically, and diagonally in any direction for as many squares as she wishes. Find the average (over all $64$ possible positions of the queen) of the number of squares the queen can reach from a particular square (do not count the square she stands on). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $x$ and $y$ are real numbers, determine the greatest possible value of the expression $\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.

Given a sequence of numbers $a_1, a_2, ..., a_{15}$, one can always construct a new sequence $b_1,b_2, ..., b_{15}$, where $b_i$ is equal to the number of terms in the sequence $\{a_k\}^{15}_{k=1}$ less than $a_i$ ($i = 1, 2,..., 15$). Is there a sequence $\{a_k\}^{15}_{k=1}$ for which the sequence $\{b_k\}^{15}_{k=1}$ is $$1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13 \,?$$

The number 104,060,465 is divisible by a five-digit prime number. What is that prime number?

Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).

Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$, find (a)max$S$ and (b) min$S$.

Let $f$ be a polynomial with real coefficients and suppose $f$ has no nonnegative real root. Prove that there exists a polynomial $h$ with real coefficients such that the coefficients of $fh$ are nonnegative.

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

If $m, n$, and $p$ are three different natural numbers, each between $2$ and $9$, what then are all the possible integer value(s) of the expression $\frac{m+n+p}{m+n}$?

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

Find all functions $f:\mathbb R \to \mathbb R$ such that for all $x,y\in \mathbb R$ $f(x+f(y))=2x+2f(y+1)$

The polynomial $$x^{10} + ?x^9 + ?x^8 + ... + ?x + 1$$ is written on the blackboard. Two players substitute (real) numbers instead of one of the question marks in turn. ($9$ turns total.) The first wins if the polynomial will have no real roots. Who wins?

The complex numbers $z_1$ and $z_2$ are given such that $z_1=-1+i$ and $z_2=2+4i$. Find the complex number $z_3$ such that $z_1,z_2,z_3$ are the points of an equilateral triangle. How many solutions do we have ?

Let \( P(x), Q(x) \) be non-constant real polynomials, such that for all positive integer \( m \), there exists a positive integer \( n \) satisfy \( P(m) = Q(n) \). Prove that (1) If \(\deg Q \mid \deg P\), then there exists real polynomial \( h(x) \) \( x \), satisfy \( P(x) = Q(h(x)) \) holds for all real number $x.$ (2) \(\deg Q \mid \deg P\).

[b]p1.[/b] Nine coins are placed in a row, alternating between heads and tails as follows: $H T H T H T H T H$. A legal move consists of turning over any two adjacent coins. (a) Give a sequence of legal moves that changes the configuration into $H H H H H H H H H$. (b) Prove that there is no sequence of legal moves that changes the original configuration into $T T T T T T T T T$. [b]p2.[/b] Find (with proof) all integers $k $that satisfy the equation $$\frac{k - 15}{2000}+\frac{k - 12}{2003}+\frac{k - 9}{2006}+\frac{k - 6}{2009}+\frac{k - 3}{2012} = \frac{k - 2000}{15}+\frac{k - 2003}{12}+\frac{k - 2006}{9}+\frac{k - 2009}{6}+\frac{k - 2012}{3}.$$ [b]p3.[/b] Some (not necessarily distinct) natural numbers from $1$ to $2015$ are written on $2015$ lottery tickets, with exactly one number written on each ticket. It is known that the sum of the numbers on any nonempty subset of tickets (including the set of all tickets) is not divisible by $2016$. Prove that the same number is written on all of the tickets. [b]p4.[/b] A set of points $A$ is called distance-distinct if every pair of points in $A$ has a different distance. (a) Show that for all infinite sets of points $B$ on the real line, there exists an infinite distance-distinct set A contained in $B$. (b) Show that for all infinite sets of points $B$ on the real plane, there exists an infinite distance-distinct set A contained in $B$. [b]p5.[/b] Let $ABCD$ be a (not necessarily regular) tetrahedron and consider six points $E, F, G, H, I, J$ on its edges $AB$, $BC$, $AC$, $AD$, $BD$, $CD$, respectively, such that $$|AE| \cdot |EB| = |BF| \cdot |FC| = |AG| \cdot |GC| = |AH| \cdot |HD| = |BI| \cdot |ID| = |CJ| \cdot |JD|.$$ Prove that the points $E, F, G, H, I$, and $J$ lie on the surface of a sphere. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a real number $\alpha$, a function $f$ is defined on pairs of nonnegative integers by $f(0,0) = 1, f(m,0) = f(0,m) = 0$ for $m > 0$, $f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$ for $m,n > 0$. Find the values of $\alpha$ such that $| f(m,n)| < 1989$ holds for any integers $m,n \ge 0$.

Let $\triangle ABC$ satisfies $\cos A:\cos B:\cos C=1:1:2$, then $\sin A=\sqrt[s]{t}$($s\in\mathbb{N},t\in\mathbb{Q^+}$ and $t$ is an irreducible fraction). Find $s+t$.

[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? [b]b.)[/b] For which $n>2$ is there exactly one set having this property?

Is there exist a function $f:\mathbb {N}\to \mathbb {N}$ with for $\forall m,n \in \mathbb {N}$ $$f\left(mf\left(n\right)\right)=f\left(m\right)f\left(m+n\right)+n ?$$

The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.

Prove that if for any positive $p$ all roots of the equation $ax^2 + bx + c + p = 0$ are real and positive then $a = 0$.

For a prime $ p\ge 5, $ determine the number of polynomials $ X^p+pX^k+pX^l+1 $ with $ 1<k<l<p, $ that are ireducible over the integers.

Are there such natural $n$, that exist polynomial of degree $n$ and with $n$ different real roots, and a) $P(x)P(x+1)=P(x^2)$ b) $P(x)P(x+1)=P(x^2+1)$