Let $\left(a_k\right)_{k=1}^\infty$ be a sequence of positive numbers such that \[a_{k-1}a_{k+1}\ge a_k^2\] for all $k>1.$ For $n\ge1$ denote \[b_n=\left(a_1a_2\cdots a_n\right)^{1/n}.\] Show that also the inequality \[b_{n-1}b_{n+1}\ge b_n^2\] holds for every $n>1.$

To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be: $ \textbf{(A)}\ \text{both rational} \qquad\textbf{(B)}\ \text{both real but not rational} \qquad\textbf{(C)}\ \text{both not real}\qquad$ $\textbf{(D)}\ \text{one real, one not real}\qquad\textbf{(E)}\ \text{one real, one not real or both not real} $

Given $a$, $b$ are positive integers greater than $1$, and for every positive integer $n$, $b^{n}-1$ divides $a^{n}-1$. Define the polynomial $p_{n}(x)$ as follows: $p_0{x}=-1$, $p_{n+1}(x)=b^{n+1}(x-1)p_{n}(bx)-a(b^{n+1}-1)p_{n}(x)$, for $n \ge 0$. Prove that there exist integers $C$ and positive integer $k$ such that $p_{k}(x)=Cx^k$.

Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$.

[b]p1.[/b] The following number is the product of the divisors of $n$. $$2^63^3$$ What is $n$? [b]p2.[/b] Let a right triangle have the sides $AB =\sqrt3$, $BC =\sqrt2$, and $CA = 1$. Let $D$ be a point such that $AD = BD = 1$. Let $E$ be the point on line $BD$ that is equidistant from $D$ and $A$. Find the angle $\angle AEB$. [b]p3.[/b] There are twelve indistinguishable blackboards that are distributed to eight different schools. There must be at least one board for each school. How many ways are there of distributing the boards? [b]p4.[/b] A Nishop is a chess piece that moves like a knight on its first turn, like a bishop on its second turn, and in general like a knight on odd-numbered turns and like a bishop on even-numbered turns. A Nishop starts in the bottom-left square of a $3\times 3$-chessboard. How many ways can it travel to touch each square of the chessboard exactly once? [b]p5.[/b] Let a Fibonacci Spiral be a spiral constructed by the addition of quarter-circles of radius $n$, where each $n$ is a term of the Fibonacci series: $$1, 1, 2, 3, 5, 8,...$$ (Each term in this series is the sum of the two terms that precede it.) What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area $714$, whose side lengths are terms in the Fibonacci series? [b]p6.[/b] Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{2}{n + 2}+\frac{4}{n + 1}-\frac{2}{n}$$ What is $a_{15}$? [b]p7.[/b] Consider $5$ points in the plane, no three of which are collinear. Let $n$ be the number of circles that can be drawn through at least three of the points. What are the possible values of $n$? [b]p8.[/b] Find the number of positive integers $n$ satisfying $\lfloor n /2014 \rfloor =\lfloor n/2016 \rfloor$. [b]p9.[/b] Let $f$ be a function taking real numbers to real numbers such that for all reals $x \ne 0, 1$, we have $$f(x) + f \left( \frac{1}{1 - x}\right)= (2x - 1)^2 + f\left( 1 -\frac{1}{ x}\right)$$ Compute $f(3)$. [b]p10.[/b] Alice and Bob split $5$ beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy? [b]p11.[/b] Triangle $ABC$ is an equilateral triangle of side length $1$. Let point $M$ be the midpoint of side $AC$. Another equilateral triangle $DEF$, also of side length $1$, is drawn such that the circumcenter of $DEF$ is $M$, point $D$ rests on side $AB$. The length of $AD$ is of the form $\frac{a+\sqrt{b}}{c}$ , where $b$ is square free. What is $a + b + c$? [b]p12.[/b] Consider the function $f(x) = \max \{-11x- 37, x - 1, 9x + 3\}$ defined for all real $x$. Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with x values $t_1$, $t_2$ and $t_3$ Compute the maximum value of $t_1 + t_2 + t_3$ over all possible $p$. [b]p13.[/b] Circle $J_1$ of radius $77$ is centered at point $X$ and circle $J_2$ of radius $39$ is centered at point $Y$. Point $A$ lies on $J1$ and on line $XY$ , such that A and Y are on opposite sides of $X$. $\Omega$ is the unique circle simultaneously tangent to the tangent segments from point $A$ to $J_2$ and internally tangent to $J_1$. If $XY = 157$, what is the radius of $\Omega$ ? [b]p14.[/b] Find the smallest positive integer $n$ so that for any integers $a_1, a_2,..., a_{527}$,the number $$\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)$$ is divisible by $527$. [b]p15.[/b] A circle $\Omega$ of unit radius is inscribed in the quadrilateral $ABCD$. Let circle $\omega_A$ be the unique circle of radius $r_A$ externally tangent to $\Omega$, and also tangent to segments $AB$ and $DA$. Similarly define circles $\omega_B$, $\omega_C$, and $\omega_D$ and radii $r_B$, $r_C$, and $r_D$. Compute the smallest positive real $\lambda$ so that $r_C < \lambda$ over all such configurations with $r_A > r_B > r_C > r_D$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f : \mathbb R\to\mathbb R $ such that \[f(x+yf(x^2))=f(x)+xf(xy)\] for all real numbers $x$ and $y$.

Let $a_n = 1 + \frac{2}{n} - \frac{2}{n^3} - \frac{1}{n^4}$. For which smallest positive integer $n$ does the value of $P_n = a_2a_3a_4 \ldots a_n$ exceed $100$?

Let $R[x]$ be the whole set of real coefficient polynomials, and define the mapping $T: R[x] \to R[x]$ as follows: For $$f (x) = a_nx^{n} + a_{n-1}x^{n- 1} +...+ a_1x + a_0,$$ let $$T(f(x))=a_{n}x^{n+1} + a_{n-1}x^{n} + (a_n+a_{n-2})x^{n-1 } + (a_{n-1}+a_{n-3})x^{n-2}+...+(a_2+a_0)x+a_1.$$ Assume $P_0(x)= 1$, $P_n(x) = T(P_{n-1}(x))$ ( $n=1,2,...$), find the constant term of $P_n(x)$.

For some constant $k$ the polynomial $p(x) = 3x^2 + kx + 117$ has the property that $p(1) = p(10)$. Evaluate $p(20)$.

Determine the real number $a$ having the property that $f(a)=a$ is a relative minimum of $f(x)=x^4-x^3-x^2+ax+1$.

[b]p1.[/b] Consider the system of simultaneous equations $$(x+y)(x+z)=a^2$$ $$(x+y)(y+z)=b^2$$ $$(x+z)(y+z)=c^2$$ , where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$. [b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ . [img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img] [b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer. [b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles. [img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img] [b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.

Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called [i]good[/i] if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$. Denote by $\mathcal{J}$ the set of polynomials of the form \[p_1q_1+\cdots+p_mq_m,\]where $m$ is a positive integer and $q_1,\ldots , q_m$ are polynomials with integer coefficients, and $p_1,\ldots , p_m$ are good polynomials. Find the smallest natural number $D{}$ such that each monomial of degree $D{}$ lies in the set $\mathcal{J}$.

Find all such functions $f :\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following holds : \[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]

Determine all real tuples $\left(x_1,x_2,x_3,x_4,x_5,x_6\right)$ such that \begin{align*} x_1(x_6 + x_2) &= x_3 + x_5, \\ x_2(x_1 + x_3) &= x_4 + x_6, \\ x_3(x_2 + x_4) &= x_5 + x_1, \\ x_4(x_3 + x_5) &= x_6 + x_2, \\ x_5(x_4 + x_6) &= x_1 + x_3, \\ x_6(x_5 + x_1) &= x_2 + x_4. \end{align*}

Pete chooses $ 1004$ monic quadratic polynomial $ f_{1},\cdots,f_{1004}$, such that each integer from $ 0$ to $ 2007$ is a root of at least one of them. Vasya considers all equations of the form $ f_{i}\equal{}f_{j}(i\not \equal{}j)$ and computes their roots; for each such root , Pete has to pay to Vasya $ 1$ ruble . Find the least possible value of Vasya's income.

[b]p1.[/b] If $x$ is $20\%$ of $23$ and $y$ is $23\%$ of $20$, compute $xy$ . [b]p2.[/b] Pablo wants to eat a banana, a mango, and a tangerine, one at a time. How many ways can he choose the order to eat the three fruits? [b]p3.[/b] Let $a$, $b$, and $c$ be $3$ positive integers. If $a + \frac{b}{c} = \frac{11}{6}$ , what is the minimum value of $a + b + c$? [b]p4.[/b] A rectangle has an area of $12$. If all of its sidelengths are increased by $2$, its area becomes $32$. What is the perimeter of the original rectangle? [b]p5.[/b] Rohit is trying to build a $3$-dimensional model by using several cubes of the same size. The model’s front view and top view are shown below. Suppose that every cube on the upper layer is directly above a cube on the lower layer and the rotations are considered distinct. Compute the total number of different ways to form this model. [img]https://cdn.artofproblemsolving.com/attachments/b/b/40615b956f3d18313717259b12fcd6efb74cf8.png[/img] [b]p6.[/b] Priscilla has three octagonal prisms and two cubes, none of which are touching each other. If she chooses a face from these five objects in an independent and uniformly random manner, what is the probability the chosen face belongs to a cube? (One octagonal prism and cube are shown below.) [img]https://cdn.artofproblemsolving.com/attachments/0/0/b4f56a381c400cae715e70acde2cdb315ee0e0.png[/img] [b]p7.[/b] Let triangle $\vartriangle ABC$ and triangle $\vartriangle DEF$ be two congruent isosceles right triangles where line segments $\overline{AC}$ and $\overline{DF}$ are their respective hypotenuses. Connecting a line segment $\overline{CF}$ gives us a square $ACFD$ but with missing line segments $\overline{AC}$, $\overline{AD}$, and $\overline{DF}$. Instead, $A$ and $D$ are connected by an arc defined by the semicircle with endpoints $A$ and $D$. If $CF = 1$, what is the perimeter of the whole shape $ABCFED$ ? [img]https://cdn.artofproblemsolving.com/attachments/2/5/098d4f58fee1b3041df23ba16557ed93ee9f5b.png[/img] [b]p8.[/b] There are two moles that live underground, and there are five circular holes that the moles can hop out of. The five holes are positioned as shown in the diagram below, where $A$, $B$, $C$, $D$, and $E$ are the centers of the circles, $AE = 30$ cm, and congruent triangles $\vartriangle ABC$, $\vartriangle CBD$, and $\vartriangle CDE$ are equilateral. The two moles randomly choose exactly two of the five holes, hop out of the two chosen holes, and hop back in. What is the probability that the holes that the two moles hop out of have centers that are exactly $15$ cm apart? [img]https://cdn.artofproblemsolving.com/attachments/c/e/b46ba87b954a1904043020d7a211477caf321d.png[/img] [b]p9.[/b] Carson is planning a trip for $n$ people. Let $x$ be the number of cars that will be used and $y$ be the number of people per car. What is the smallest value of $n$ such that there are exactly $3$ possibilities for $x$ and $y$ so that $y$ is an integer, $x < y$, and exactly one person is left without a car? [b]p10.[/b] Iris is eating an ice cream cone, which consists of a hemisphere of ice cream with radius $r > 0$ on top of a cone with height $12$ and also radius $r$. Iris is a slow eater, so after eating one-third of the ice cream, she notices that the rest of the ice cream has melted and completely filled the cone. Assuming the ice cream did not change volume after it melted, what is the value of $r$? [b]p11.[/b] As Natasha begins eating brunch between $11:30$ AM and $12$ PM, she notes that the smaller angle between the minute and hour hand of the clock is $27$ degrees. What is the number of degrees in the smaller angle between the minute and hour hand when Natasha finishes eating brunch $20$ minutes later? [b]p12.[/b] On a regular hexagon $ABCDEF$, Luke the frog starts at point $A$, there is food on points $C$ and $E$ and there are crocodiles on points $B$ and $D$. When Luke is on a point, he hops to any of the five other vertices with equal probability. What is the probability that Luke will visit both of the points with food before visiting any of the crocodiles? [b]p13.[/b] $2023$ regular unit hexagons are arranged in a tessellating lattice, as follows. The first hexagon $ABCDEF$ (with vertices in clockwise order) has leftmost vertex $A$ at the origin, and hexagons $H_2$ and $H_3$ share edges $\overline{CD}$ and $\overline{DE}$ with hexagon $H_1$, respectively. Hexagon $H_4$ shares edges with both hexagons $H_2$ and $H_3$, and hexagons $H_5$ and $H_6$ are constructed similarly to hexagons H_2 and $H_3$. Hexagons $H_7$ to $H_{2022}$ are constructed following the pattern of hexagons $H_4$, $H_5$, $H_6$. Finally, hexagon H_{2023} is constructed, sharing an edge with both hexagons H2021 and H2022. Compute the perimeter of the resulting figure. [img]https://cdn.artofproblemsolving.com/attachments/1/d/eaf0d04676bac3e3c197b4686dcddd08fce9ac.png[/img] [b]p14.[/b] Aditya’s favorite number is a positive two-digit integer. Aditya sums the integers from $5$ to his favorite number, inclusive. Then, he sums the next $12$ consecutive integers starting after his favorite number. If the two sums are consecutive integers and the second sum is greater than the first sum, what is Aditya’s favorite number? [b]p15.[/b] The $100^{th}$ anniversary of BMT will fall in the year $2112$, which is a palindromic year. Compute the sum of all years from $0000$ to $9999$, inclusive, that are palindromic when written out as four-digit numbers (including leading zeros). Examples include $2002$, $1991$, and $0110$. [b]p16.[/b] Points $A$, $B$, $C$, $D$, and $E$ lie on line $r$, in that order, such that $DE = 2DC$ and $AB = 2BC$. Let $M$ be the midpoint of segment $\overline{AC}$. Finally, let point $P$ lie on $r$ such that $PE = x$. If $AB = 8x$, $ME = 9x$, and $AP = 112$, compute the sum of the two possible values of $CD$. [b]p17.[/b] A parabola $y = x^2$ in the xy-plane is rotated $180^o$ about a point $(a, b)$. The resulting parabola has roots at $x = 40$ and $x = 48$. Compute $a + b$. [b]p18.[/b] Susan has a standard die with values $1$ to $6$. She plays a game where every time she rolls the die, she permanently increases the value on the top face by $1$. What is the probability that, after she rolls her die 3 times, there is a face on it with a value of at least $7$? [b]p19.[/b] Let $N$ be a $6$-digit number satisfying the property that the average value of the digits of $N^4$ is $5$. Compute the sum of the digits of $N^4$. [b]p20.[/b] Let $O_1$, $O_2$, $...$, $O_8$ be circles of radius $1$ such that $O_1$ is externally tangent to $O_8$ and $O_2$ but no other circles, $O_2$ is externally tangent to $O_1$ and $O_3$ but no other circles, and so on. Let $C$ be a circle that is externally tangent to each of $O_1$, $O_2$, $...$, $O_8$. Compute the radius of $C$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence? (Alexandr Shapovalov)

Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.

Let $n$ be an odd number larger than 1, and $f(x)$ is a polynomial with degree $n$ such that $f(k)=2^k$ for $k=0,1,\cdots,n$. Prove that there is only finite integer $x$ such that $f(x)$ is the power of two.

Let $a,b,c$ three positive numbers less than $ 1$. Prove that cannot occur simultaneously these three inequalities: $$a (1- b)>\frac14$$ $$b (1-c)>\frac14 $$ $$c (1-a)>\frac14$$

Let $x,y,z \in \mathbb{R}^{+}$ and $x^2+y^2+z^2=x+y+z$. Prove that $$x+y+z+3 \ge 6 \sqrt[3]{\frac{xy+yz+zx}{3}}$$

On the coordinate plane, the parabola $y = x^2$ and the points $A(x_1, x_1^2)$, $B(x_2, x_2^2)$ are set such that $x_1=-998$, $x_2 =1999$ The segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$ and $X_k$ are constructed succesively with $(x_k,0)$, $1 \le k \le 1998$ and $x_3$, $x_4$,..., $x_{1998}$ are abscissas of the points of intersection of the parabola with segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$. Find the value $\frac{1}{x_{1999}}+\frac{1}{x_{2000}}$

Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

If $x, y, z$ are real numbers satisfying $x^2 +y^2 +z^2 = 2$, prove the inequality \[ x + y + z \leq 2 + xyz \] When does equality occur?