For how many real numbers $a$ does the quadratic equation $x^2 + ax + 6a=0$ have only integer roots for $x$?

The sum of positive real numbers $ a,b,c,d$ is $ 1$. Prove that: $ \frac{a^2}{a\plus{}b}\plus{}\frac{b^2}{b\plus{}c}\plus{}\frac{c^2}{c\plus{}d}\plus{}\frac{d^2}{d\plus{}a} \ge \frac{1}{2},$ with equality if and only if $ a\equal{}b\equal{}c\equal{}d\equal{}\frac{1}{4}$.

In a scalene triangle $ABC$, $D$ is the foot of the altitude through $A$, $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$. Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$, $Y=BE\cap CF$, $Z=CF \cap AD$. If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$, $OYZ$, $OZX$ must be equilateral.

Suppose that $n$ and $k$ are positive integers such that \[ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). \] Prove that $n \le 3^k$. Here $\varphi(n)$ denotes Euler's totient function, i.e. $\varphi(n)$ denotes the number of elements of $\{1, \dots, n\}$ which are relatively prime to $n$. In particular, $\varphi(1) = 1$. [i]Proposed by Linus Hamilton[/i]

Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.

In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.

You are given $2n$ distinct integers. What's the largest integer $C$ such that you can always form at least $C$ pairs from them, so that no integer is in more than one pair, and the sum of integers in each pair is a composite number? [i](Proposed by Anton Trygub)[/i]

Prove that for every real positive number $a, b, c$ with $1 \le a, b, c \le 8$ the inequality $$\frac{a+b+c}{5}\le \sqrt[3]{abc}$$

$A$ gives $B$ as many cents as $B$ has and $C$ as many cents as $C$ has. Similarly, $B$ then gives $A$ and $C$ as many cents as each then has. $C$, similarly, then gives $A$ and $B$ as many cents as each then has. If each finally has $16$ cents, with how many cents does $A$ start? $\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26\qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\geq 0}$ satisfying the equation \[a_{n+1}=x-\frac{1}{a_{n}}\quad (n\geq 0).\] A positive integer $p$ is called a [i]minimal period[/i] of the family $F(x)$ if (a) each sequence $\left(a_{n}\right)\in F(x)$ is periodic with the period $p$, (b) for each $0<q<p$ there exists $\left(a_{n}\right)\in F(x)$ such that $q$ is not a period of $\left(a_{n}\right)$. Prove or disprove that for each positive integer $P$ there exists a real number $x=x(P)$ such that the family $F(x)$ has the minimal period $p>P$.

Find all positive integers $k$ such that there exists some permutation of $(1, 2,...,1000)$ namely $(a_1, a_2,..., a_{1000}) $ and satisfy $|a_i - i| = k$ for all $i = 1,1000$.

Let $a$ be an odd natural number and $b$ be a positive integer. We define a sequence of reals $(u_n)$ as follows: $u_0=b$ and, for all $n\in\mathbb N_0$, $u_{n+1}$ is $\frac{u_n}2$ if $u_n$ is even and $a+u_n$ otherwise. (a) Prove that one can find an element of $u_n$ smaller than $a$. (b) Prove that the sequence is eventually periodic.

[b]p1.[/b] Farmer James goes to Kristy’s Krispy Chicken to order a crispy chicken sandwich. He can choose from $3$ types of buns, $2$ types of sauces, $4$ types of vegetables, and $4$ types of cheese. He can only choose one type of bun and cheese, but can choose any nonzero number of sauces, and the same with vegetables. How many different chicken sandwiches can Farmer James order? [b]p2.[/b] A line with slope $2$ and a line with slope $3$ intersect at the point $(m, n)$, where $m, n > 0$. These lines intersect the $x$ axis at points $A$ and $B$, and they intersect the y axis at points $C$ and $D$. If $AB = CD$, find $m/n$. [b]p3.[/b] A multi-set of $11$ positive integers has a median of $10$, a unique mode of $11$, and a mean of $ 12$. What is the largest possible number that can be in this multi-set? (A multi-set is a set that allows repeated elements.) [b]p4.[/b] Farmer James is swimming in the Eggs-Eater River, which flows at a constant rate of $5$ miles per hour, and is recording his time. He swims $ 1$ mile upstream, against the current, and then swims $1$ mile back to his starting point, along with the current. The time he recorded was double the time that he would have recorded if he had swum in still water the entire trip. To the nearest integer, how fast can Farmer James swim in still water, in miles per hour? [b]p5.[/b] $ABCD$ is a square with side length $60$. Point $E$ is on $AD$ and $F$ is on $CD$ such that $\angle BEF = 90^o$. Find the minimum possible length of $CF$. [b]p6.[/b] Farmer James makes a trianglomino by gluing together $5$ equilateral triangles of side length $ 1$, with adjacent triangles sharing an entire edge. Two trianglominoes are considered the same if they can be matched using only translations and rotations (but not reflections). How many distinct trianglominoes can Farmer James make? [b]p7.[/b] Two real numbers $x$ and $y$ satisfy $x^2 - y^2 = 2y - 2x$ , and $x + 6 = y^2 + 2y$. What is the sum of all possible values of$ y$? [b]p8.[/b] Let $N$ be a positive multiple of $840$. When $N$ is written in base $6$, it is of the form $\overline{abcdef}_6$ where $a, b, c, d, e, f$ are distinct base $6$ digits. What is the smallest possible value of $N$, when written in base $6$? [b]p9.[/b] For $S = \{1, 2,..., 12\}$, find the number of functions $f : S \to S$ that satisfy the following $3$ conditions: (a) If $n$ is divisible by $3$, $f(n)$ is not divisible by $3$, (b) If $n$ is not divisible by $3$, $f(n)$ is divisible by $3$, and (c) $f(f(n)) = n$ holds for exactly $8$ distinct values of $n$ in $S$. [b]p10.[/b] Regular pentagon $JAMES$ has area $ 1$. Let $O$ lie on line $EM$ and $N$ lie on line $MA$ so that $E, M, O$ and $M, A, N$ lie on their respective lines in that order. Given that $MO = AN$ and $NO = 11 \cdot ME$, find the area of $NOM$. [b]p11.[/b] Hen Hao is flipping a special coin, which lands on its sunny side and its rainy side each with probability $1/2$. Hen Hao flips her coin ten times. Given that the coin never landed with its rainy side up twice in a row, find the probability that Hen Hao’s last flip had its sunny side up. [b]p12.[/b] Find the product of all integer values of a such that the polynomial $x^4 + 8x^3 + ax^2 + 2x - 1$ can be factored into two non-constant polynomials with integer coefficients. [b]p13.[/b] Isosceles trapezoid $ABCD$ has $AB = CD$ and $AD = 6BC$. Point $X$ is the intersection of the diagonals $AC$ and $BD$. There exist a positive real number $k$ and a point $P$ inside $ABCD$ which satisfy $$[PBC] : [PCD] : [PDA] = 1 : k : 3,$$ where $[XYZ]$ denotes the area of triangle $XYZ$. If $PX \parallel AB$, find the value of $k$. [b]p14.[/b] How many positive integers $n < 1000$ are there such that in base $10$, every digit in $3n$ (that isn’t a leading zero) is greater than the corresponding place value digit (possibly a leading zero) in $n$? For example, $n = 56$, $3n = 168$ satisfies this property as $1 > 0$, $6 > 5$, and $8 > 6$. On the other hand, $n = 506$, $3n = 1518$ does not work because of the hundreds place. [b]p15.[/b] Find the greatest integer that is smaller than $$\frac{2018}{37^2}+\frac{2018}{39^2}+ ... +\frac{2018}{ 107^2}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $ a,b,c$ are positive integers less than $ 10$, then $ (10a \plus{} b)(10a \plus{} c) \equal{} 100a(a \plus{} 1) \plus{} bc$ if: $ \textbf{(A)}\ b \plus{} c \equal{} 10 \qquad\textbf{(B)}\ b \equal{} c \qquad\textbf{(C)}\ a \plus{} b \equal{} 10 \qquad\textbf{(D)}\ a \equal{} b \\ \textbf{(E)}\ a \plus{} b \plus{} c \equal{} 10$

Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m)=20$ and $s(33m)=120$, what is the value of $s(3m)$?

Triangle $ABC$ is right-angled and isosceles with a right angle at the vertex $C$. On rays $CB$ on vertex $B$ is selected point F, on rays $BA$ on vertex $A$ is selected point G so that $AG = BF.$ The ray $GD$ is drawn so that it intersects with ray $AC$ at point $D$ with $\angle FGD = 45^o$. Find $\angle FDG$. (Bogdan Rublev)

Let $C(n)$ be the number of prime divisors of a positive integer $n$. (a) Consider set $S$ of all pairs of positive integers $(a, b)$ such that $a \ne b$ and $C(a + b) = C(a) + C(b)$. Is $S$ finite or infinite? (b) Define $S'$ as a subset of S consisting of the pairs $(a, b)$ such that $C(a+b) > 1000$. Is $S'$ finite or infinite?

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$. [i]Bulgaria[/i]

There is a unique continuous function $f$ over the positive real numbers satisfying $f(4) = 1$ and \[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \] for all positive $x$. Compute the value of $\int_{0}^{140} (f(x))^3\,dx$.

Let $ a $ be a positive real number. Calculate $ \lim_{n\to\infty} \frac{a^n}{(1+a)(1+a^2)\cdots (1+a^n)} . $

Let $a,b,c>0.$ Prove that $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \ge \frac{1}{\sqrt{2a^2+2bc}}+\frac{1}{\sqrt{2b^2+2ca}}+\frac{1}{\sqrt{2c^2+2ab}}$

Determine the largest integer $k$ for which $12^k$ is a factor of $120! $

The number of triangles such that lengths of three sides are integers, and the length of the longest side is $11$ is________.

Let $R$ be the set of points $(x, y)$ such that $x$ and $y$ are positive, $x + y$ is at most 2013, and \[ \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil. \] Compute the area of set $R$. Recall that $\lfloor a \rfloor$ is the greatest integer that is less than or equal to $a$, and $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.

There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a [i]match[/i] be cause they have distinct amount of shapes, distinct shapes but the same color of shapes. What is the maximum number of cards that we can choose such that non of the triples make a [i]match[/i]? [i]Proposed by Amin Bahjati[/i]