Let $P(x,y)=x^2y+xy^2$ and $Q(x,y)=x^2+xy+y^2.$ For $n=1,2,3,\dots,$ let \begin{align*}F_n(x,y)=(x+y)^n-x^n-y^n \text{ and,}\\ G_n(x,y)=(x+y)^n+x^n+y^n. \end{align*} One observes that $$G_2=2Q, F_3=3P, G_4=2Q^2, F_5=5PQ, G_6=2Q^3+3P^2.$$ Prove that, in fact, for each $n$ either $F_n$ or $G_n$ is expressible as a polynomial in $P$ and $Q$ with integer coefficients.

Find all pairs of nonnegative integers $ (x, y)$ satisfying $ (14y)^x \plus{} y^{x\plus{}y} \equal{} 2007$.

In the year $ 2001$, the United States will host the International Mathematical Olympiad. Let $ I$, $ M$, and $ O$ be distinct positive integers such that the product $ I\cdot M \cdot O \equal{} 2001$. What's the largest possible value of the sum $ I \plus{} M \plus{} O$? $ \textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ 111 \qquad \textbf{(E)}\ 671$

\[\int_e^{e^2} (\log(x))^{\log(x)}(2+\log(\log(x)))\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.

Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? [hide=Solution] With probability $1$ the number of corners covered is $0$, $1$, or $2$ for example by the diameter of a square being $\sqrt{2}$ so it suffices to compute the probability that the square covers $2$ corners. This is due to the fact that density implies the mean number of captured corners is $1$. For the lattice with offset angle $\theta \in \left[0,\frac{\pi}{2}\right]$ consider placing a lattice uniformly randomly on to it and in particular say without loss of generality consider the square which covers the horizontal lattice midpoint $\left(\frac{1}{2},0 \right)$. The locus of such midpoint locations so that the square captures the $2$ points $(0,0),(1,0)$, is a rectangle. As capturing horizontally adjacent points does not occur when capturing vertically adjacent points one computes twice that probability as $\frac{4}{\pi} \int_0^{\frac{\pi}{2}} (1-\sin(\theta))(1-\cos(\theta)) d\theta=\boxed{\frac{2(\pi-3)}{\pi}}$ \\ [asy] draw((0,0)--(80,40)); draw((0,0)--(-40,80)); draw((80,40)--(40,120)); draw((-40,80)--(40,120)); draw((80,40)--(-20,40)); draw((-40,80)--(60,80)); draw((32*sqrt(5),16*sqrt(5))--(-8*sqrt(5),16*sqrt(5))); draw((40+8*sqrt(5),120-16*sqrt(5))--(40-32*sqrt(5),120-16*sqrt(5))); draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)+2*(40-16*sqrt(5)),16*sqrt(5)+(40-16*sqrt(5)))); draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)-(80-16*sqrt(5))/2,16*sqrt(5)+(80-16*sqrt(5)))); draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)+(120-16*sqrt(5)-40)/2,120-16*sqrt(5)-(120-16*sqrt(5)-40))); draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)-2*(120-16*sqrt(5)-80),120-16*sqrt(5)-(120-16*sqrt(5)-80))); [/asy] [/hide]

Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$f(ac)+f(bc)-f(c)f(ab) \ge 1$$ Proposed by [i]Mojtaba Zare[/i]

Find all positive integers $n$ such that the number $A_n =\frac{ 2^{4n+2}+1}{65}$ is a) an integer, b) a prime.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Find all real numbers $ x$ which satisfy the following equation: $ [2x]\plus{}[3x]\plus{}[7x]\equal{}2008$. Note: $ [x]$ means the greatest integer less or equal than $ x$.

Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.

Let n be a positive integer and let $x_1, x_2,...,x_n$ and $y_1, y_2,...,y_n$ be real numbers. Prove that there exists a number $i, i = 1, 2,...,n$, such that $$\sum_{j=1}^n |x_i - x_j | \le \sum_{j=1}^n |x_i - y_j | $$

Positive integers $ a,b,$ and $ c$ are randomly and independently selected with replacement from the set $ \{ 1,2,3,\dots,2010 \}.$ What is the probability that $ abc \plus{} ab \plus{} a$ is divisible by $ 3$? $ \textbf{(A)}\ \dfrac{1}{3} \qquad\textbf{(B)}\ \dfrac{29}{81} \qquad\textbf{(C)}\ \dfrac{31}{81} \qquad\textbf{(D)}\ \dfrac{11}{27} \qquad\textbf{(E)}\ \dfrac{13}{27}$

There are $N$ points in the plane such that the [b]total number[/b] of pairwise distances of these $N$ points is at most $n$. Prove that $N \le (n + 1)^2$.

Pyramid $EARLY$ has rectangular base $EARL$ and apex $Y$, and all of its edges are of integer length. The four edges from the apex have lengths $1, 4, 7, 8$ (in no particular order), and EY is perpendicular to $YR$. Find the area of rectangle $EARL$.

Let $\mathbb{S}$ is the set of prime numbers that less or equal to 26. Is there any $a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}$ such that $$ gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6$$ and for every element $p$ of $\mathbb{S}$ there exists a pair of $ 1\leq k \ne l \leq 6$ such that $$s=gcd(a_k,a_l)?$$

Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that $$\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$$

Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$

There are $n$ ordered tuples of positive integers $(a,b,c,d)$ that satisfy $$a^2+ b^2+ c^2+ d^2=13 \cdot 2^{13}.$$ Let these ordered tuples be $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2), \dots, (a_n,b_n,c_n,d_n)$. Compute $\sum_{i=1}^{n}(a_i+b_i+c_i+d_i)$. [i]Proposed by Kaylee Ji[/i]

A monic quadratic polynomial $f$ with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.

Let $ ABCD$ be a quadrilateral inscribed in a circle. The diagonal $ BD$ bisects $ AC$. If $ AB = 10$, $ AD = 12$ and $ DC = 11$, find $ BC$.

Let $a, b, c, d$ be real numbers satisfying the system of equation $\[(a+b)(c+d)=2 \\ (a+c)(b+d)=3 \\ (a+d)(b+c)=4\]$ Find the minimum value of $a^2+b^2+c^2+d^2$.

Find the number of ordered pairs $(x,y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x\times y$ ends in the digit $5$. [i]Proposed by Andrew Wen[/i]

[b]8.1[/b] Four circles are placed on planes so that each one touches the other two externally. Prove that the points of tangency lie on one circle. [img]https://cdn.artofproblemsolving.com/attachments/9/8/883a82fb568954b09a4499a955372e2492dbb8.png[/img] [b]8.2[/b]. Let the integers $a$ and $b$ be represented as $x^2-5y^2$, where $x$ and $y$ are integer numbers. Prove that the number $ab$ can also be presented in this form. [b]8.3[/b] Solve the equation $x(x + d)(x + 2d)(x + 3d) = a$. [b]8.4 / 9.1[/b] Let $a+b+c=1$, $m+n+p=1 $. Prove that $$-1 \le am + bn + cp \le 1 $$ [b]8.5[/b] Inscribe a triangle with the largest area in a semicircle. [b]8.6[/b] Three circles of the same radius intersect at one point. Prove that the other three points intersections lie on a circle of the same radius. [img]https://cdn.artofproblemsolving.com/attachments/4/7/014952f2dcf0349d54b07230e45a42c242a49d.png[/img] [b]8.7[/b] Find the circle of smallest radius that contains a given triangle. [b]8.8 / 9.2[/b] Given a polynomial $$x^{2n} +a_1x^{2n-2} + a_2x^{2n-4} + ... + a_{n-1}x^2 + a_n,$$ which is divisible by $ x-1$. Prove that it is divisible by $x^2-1$. [b]8.9[/b] Prove that for any prime number $p$ other than $2$ and from $5$, there is a natural number $k$ such that only ones are involved in the decimal notation of the number $pk$.. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].