Given the nonincreasing sequence of non-negative numbers in which $a_1 \ge a_2 \ge a_3 \ge ... \ge a_{2n-1}\ge 0$. Prove that $a^2_1 -a^2_2 + a^2_3 - ... + a^2_{2n- l} \ge (a_1 - a_2 + a_3 - ... + a_{2n- l} )^2$ . ( L . Kurlyandchik , Leningrad )

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]

if three non-zero vectors on a plane $\vec{a},\vec{b},\vec{c}$ satisfy: (a) $\vec{a}\bot\vec{b}$ (b) $\vec{b}\cdot\vec{c}=2|\vec{a}|$ (c) $\vec{c}\cdot\vec{a}=3|\vec{b}|$ find out the minimum of $|\vec{c}|$

Let two circles $A$ and $B$ with unequal radii $r$ and $R$, respectively, be tangent internally at the point $A_0$. If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$, and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$, prove that \[\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.\]

Find all ordered pairs of real numbers $(x, y)$ such that $x^2y = 3$ and $x + xy = 4$.

The number $A$ is a product of $n$ distinct natural numbers. Prove that $A$ has at least $\frac{n(n-1)}{2}+1$ distinct divisors (including 1 and $A$).

Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length? $ \textbf{(A) }\frac25\qquad\textbf{(B) }\frac49\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac59\qquad\textbf{(E) }\frac45 $

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute $2^2 + 0 \cdot 0 + 2^2 + 3^3$. [b]p2.[/b] How many total letters (not necessarily distinct) are there in the names Jerry, Justin, Jackie, Jason, and Jeffrey? [b]p3.[/b] What is the remainder when $20232023$ is divided by $50$? [b]p4.[/b] Let $ABCD$ be a square. The fraction of the area of $ABCD$ that is the area of the intersection of triangles $ABD$ and $ABC$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$. [b]p5.[/b] Raymond is playing basketball. He makes a total of $15$ shots, all of which are either worth $2$ or $3$ points. Given he scored a total of $40$ points, how many $2$-point shots did he make? [b]p6.[/b] If a fair coin is flipped $4$ times, the probability that it lands on heads more often than tails is $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$. [b]p7.[/b] What is the sum of the perfect square divisors of $640$? [b]p8.[/b] A regular hexagon and an equilateral triangle have the same perimeter. The ratio of the area between the hexagon and equilateral triangle can be expressed in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p9.[/b] If a cylinder has volume $1024\pi$, radius of $r$ and height $h$, how many ordered pairs of integers $(r, h)$ are possible? [b]p10.[/b] Pump $A$ can fill up a balloon in $3$ hours, while pump $B$ can fill up a balloon in $5$ hours. Pump $A$ starts filling up a balloon at $12:00$ PM, and pump $B$ is added alongside pump $A$ at a later time. If the balloon is completely filled at $2:00$ PM, how many minutes after $12:00$ PM was Pump $B$ added? [b]p11.[/b] For some positive integer $k$, the product $81 \cdot k$ has $20$ factors. Find the smallest possible value of $k$. [b]p12.[/b] Two people wish to sit in a row of fifty chairs. How many ways can they sit in the chairs if they do not want to sit directly next to each other and they do not want to sit with exactly one empty chair between them? [b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $2$ and $M$ be the midpoint of $BC$. Let $P$ be a point in the same plane such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}-b$, where $a$ and $b$ are positive integers such that $a$ is not divisible by any perfect square aside from $1$. Find $a + b$. [b]p14.[/b] What are the $2022$nd to $2024$th digits after the decimal point in the decimal expansion of $\frac{1}{27}$ , expressed as a $3$ digit number in that order (i.e the $2022$nd digit is the hundreds digit, $2023$rd digit is the tens digit, and $2024$th digit is the ones digit)? [b]p15.[/b] After combining like terms, how many terms are in the expansion of $(xyz+xy+yz+xz+x+y+z)^{20}$? [b]p16.[/b] Let $ABCD$ be a trapezoid with $AB \parallel CD$ where $AB > CD$, $\angle B = 90^o$, and $BC = 12$. A line $k$ is dropped from $A$, perpendicular to line $CD$, and another line $\ell$ is dropped from $C$, perpendicular to line $AD$. $k$ and $\ell$ intersect at $X$. If $\vartriangle AXC$ is an equilateral triangle, the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers such that $n$ is not divisible by any perfect square aside from $1$. Find $m + n$. [b]p17.[/b] If real numbers $x$ and $y$ satisfy $2x^2 + y^2 = 8x$, maximize the expression $x^2 + y^2 + 4x$. [b]p18.[/b] Let $f(x)$ be a monic quadratic polynomial with nonzero real coefficients. Given that the minimum value of $f(x)$ is one of the roots of $f(x)$, and that $f(2022) = 1$, there are two possible values of $f(2023)$. Find the larger of these two values. [b]p19.[/b] I am thinking of a positive integer. After realizing that it is four more than a multiple of $3$, four less than a multiple of $4$, four more than a multiple of 5, and four less than a multiple of $7$, I forgot my number. What is the smallest possible value of my number? [b]p20.[/b] How many ways can Aston, Bryan, Cindy, Daniel, and Evan occupy a row of $14$ chairs such that none of them are sitting next to each other? [b]p21.[/b] Let $x$ be a positive real number. The minimum value of $\frac{1}{x^2} +\sqrt{x}$ can be expressed in the form \frac{a}{b^{(c/d)}} , where $a$, $b$, $c$, $d$ are all positive integers, $a$ and $b$ are relatively prime, $c$ and $d$ are relatively prime, and $b$ is not divisible by any perfect square. Find $a + b + c + d$. [b]p22.[/b] For all $x > 0$, the function $f(x)$ is defined as $\lfloor x \rfloor \cdot (x + \{x\})$. There are $24$ possible $x$ such that $f(x)$ is an integer between $2000$ and $2023$, inclusive. If the sum of these $24$ numbers equals $N$, then find $\lfloor N \rfloor$. Note: Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, called the floor function. Also, $\{x\}$ is defined as $x - \lfloor x \rfloor$, called the fractional part function. [b]p23.[/b] Let $ABCD$ be a rectangle with $AD = 1$. Let $P$ be a point on diagonal $\overline{AC}$, and let $\omega$ and $\xi$ be the circumcircles of $\vartriangle APB$ and $\vartriangle CPD$, respectively. Line $\overleftrightarrow{AD}$ is extended, intersecting $\omega$ at $X$, and $\xi$ at $Y$ . If $AX = 5$ and $DY = 2$, find $[ABCD]^2$. Note: $[ABCD]$ denotes the area of the polygon $ABCD$. [b]p24.[/b] Alice writes all of the three-digit numbers on a blackboard (it’s a pretty big blackboard). Let $X_a$ be the set of three-digit numbers containing a somewhere in its representation, where a is a string of digits. (For example, $X_{12}$ would include $12$, $121$, $312$, etc.) If Bob then picks a value of $a$ at random so $0 \le a \le 999$, the expected number of elements in $X_a$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find$ m + n$. [b]p25.[/b] Let $f(x) = x^5 + 2x^4 - 2x^3 + 4x^2 + 5x + 6$ and $g(x) = x^4 - x^3 + x^2 - x + 1$. If $a$, $b$, $c$, $d$ are the roots of $g(x)$, then find $f(a) + f(b) + f(c) + f(d)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p>0$ be a real constant. From any point $(a,b)$ in the cartesian plane, show that i) Three normals, real or imaginary, can be drawn to the parabola $y^2=4px$. ii) These are real and distinct if $4(2-p)^3 +27pb^2<0$. iii) Two of them coincide if $(a,b)$ lies on the curve $27py^2=4(x-2p)^3$. iv) All three coincide only if $a=2p$ and $b=0$.

[b]Q.[/b]Let $ABC$ be a triangle with incenter $I$ and orthocenter $H$. Let $A_1,A_2$ lie on $\overline{BC}$ such that $\overline{IA_1}\perp \overline{IB},\overline{IA_2}\perp\overline{IC}$. $\overline{AA_1},\overline{AA_2}$ cut $\odot(ABC)$ again at $A_3,A_4$. $\overline{A_3A_4}$ cuts $\overline{BC}$ at $A_0$. Similarly, we have $B_0,C_0$. Prove that $A_0,B_0,C_0$ are collinear on a line which is perpendicular to line $\overline{IH}$.

Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? [i]A. Golovanov[/i]

Let $n = 2^k$. Prove that we can select $n$ integers from any $2n-1$ integers such that their sum is divisible by $n$.

The price of an article was increased $ p\%$. Later the new price was decreased $ p\%$. If the last price was one dollar, the original price was: $ \textbf{(A)}\ \frac {1 \minus{} p^2}{200} \qquad\textbf{(B)}\ \frac {\sqrt {1 \minus{} p^2}}{100} \qquad\textbf{(C)}\ \text{one dollar} \qquad\textbf{(D)}\ 1 \minus{} \frac {p^2}{10000 \minus{} p^2}$ $ \textbf{(E)}\ \frac {10000}{10000 \minus{} p^2}$

Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]

The diagonals of a trapezoid $ABCD$ with bases $AB$ and $CD$ intersect in a point $O$, and $AB/CD=k>1$. The bisectors of the angles $AOB,BOC,COD,DOA$ intersect $AB,BC,CD,DA$ respectively at $K,L,M,N$. The lines $KL$ and $MN$ meet at $P$, and the lines $KN$ and $LM$ meet at $Q$. If the areas of $ABCD$ and $OPQ$ are equal, find the value of $k$.

Inside an equilateral triangle there is a point whose distances from the sides of the triangle are $3, 4$ and $5$. Find the area of the triangle.

Given a triangle $ABC$, points $X,Y,$ and $Z$ are the midpoints of $BC,CA,$ and $AB$ respectively. The perpendicular bisector of $AB$ intersects line $XY$ and line $AC$ at $Z_1$ and $Z_2$ respectively. The perpendicular bisector of $AC$ intersects line $XZ$ and line $AB$ at $Y_1$ and $Y_2$ respectively. Let $K$ be a point such that $KZ_1 = KZ_2$ and $KY_1 = KY_2$. Prove that $KB=KC$.

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(x)f(y)=f(x-y)$. Find all possible values of $f(2017)$.

There are $n$ rectangles drawn on the rectangular sheet of paper with the sides of the rectangles parallel to the sheet sides. The rectangles do not have pairwise common interior points. Prove that after cutting out the rectangles the sheet will split into not more than $n+1$ part.

Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Prove that if a polynomial with integer coefficients takes a value equal to $1$ in absolute value at three different integer points, then it has no integer zeros.

Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.

[asy] real t=pi/8;real u=7*pi/12;real v=13*pi/12; real ct=cos(t);real st=sin(t);real cu=cos(u);real su=sin(u); draw(unitcircle); draw((ct,st)--(-ct,st)--(cos(v),sin(v))); draw((cu,su)--(cu,st)); label("A",(-ct,st),W);label("B",(ct,st),E); label("M",(cu,su),N);label("P",(cu,st),S); label("C",(cos(v),sin(v)),W); //Credit to Zimbalono for the diagram[/asy] In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to $\textbf{(A) }3x+2\qquad\textbf{(B) }3x+1\qquad\textbf{(C) }2x+3\qquad\textbf{(D) }2x+2\qquad \textbf{(E) }2x+1$

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?