Let $0<a<b.$ Evaluate $$\lim_{t\to 0} \{ \int_{0}^{1} [bx+a(1-x)]^t dx\}^{1/t}.$$ [i][The final answer should not involve any other operations other than addition, subtraction, multiplication, division and exponentiation.][/i]

Let $A = (a_1, a_2, \cdots ,a_{2000})$ be a sequence of integers each lying in the interval $[-1000,1000]$. Suppose that the entries in A sum to $1$. Show that some nonempty subsequence of $A$ sums to zero.

In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD =DP = 3$, and $GC = 4$, find $AB^2$. [i]Proposed by Muztaba Syed[/i]

Consider a $2023\times2023$ board split into unit squares. Two unit squares are called adjacent is they share at least one vertex. Mahler and Srecko play a game on this board. Initially, Mahler has one piece placed on the square marked [b]M[/b], and Srecko has a piece placed on the square marked by [b]S[/b] (see the attachment). The players alternate moving their piece, following three rules: 1. A piece can only be moved to a unit square adjacent to the one it is placed on. 2. A piece cannot be placed on a unit square on which a piece has been placed before (once used, a unit square can never be used again). 3. A piece cannot be moved to a unit square adjacent to the square occupied by the opponent’s piece. A player wins the game if his piece gets to the corner diagonally opposite to its starting position (i.e. Srecko moves to $s_p$, Mahler moves to $m_p$) or if the opponent has to move but has no legal move. Mahler moves first. Which player has a winning strategy?

There are $99$ space stations. Each pair of space stations is connected by a tunnel. There are $99$ two-way main tunnels, and all the other tunnels are strictly one-way tunnels. A group of $4$ space stations is called [i]connected[/i] if one can reach each station in the group from every other station in the group without using any tunnels other than the $6$ tunnels which connect them. Determine the maximum number of connected groups.

[b]p1.[/b] For $25$ bagels they paid as many rubles as the number of bagels you can buy with a ruble. How much does one bagel cost? [b]p2.[/b] Cut the square into the figure into$ 4$ parts of the same shape and size so that each part contains exactly one shaded square. [img]https://cdn.artofproblemsolving.com/attachments/a/2/14f0d435b063bcbc55d3dbdb0a24545af1defb.png[/img] [b]p3.[/b] The numerator and denominator of the fraction are positive numbers. The numerator is increased by $1$, and the denominator is increased by $10$. Can this increase the fraction? [b]p4.[/b] The brother left the house $5$ minutes later than his sister, following her, but walked one and a half times faster than her. How many minutes after leaving will the brother catch up with his sister? [b]p5.[/b] Three apples are worth more than five pears. Can five apples be cheaper than seven pears? Can seven apples be cheaper than thirteen pears? (All apples cost the same, all pears too.) [b]p6.[/b] Give an example of a natural number divisible by $6$ and having exactly $15$ different natural divisors (counting $1$ and the number itself). [b]p7.[/b] In a round dance, $30$ children stand in a circle. Every girl's right neighbor is a boy. Half of the boys have a boy on their right, and all the other boys have a girl on their right. How many boys and girls are there in a round dance? [b]p8.[/b] A sheet of paper was bent in half in a straight line and pierced with a needle in two places, and then unfolded and got $4$ holes. The positions of three of them are marked in figure Where might the fourth hole be? [img]https://cdn.artofproblemsolving.com/attachments/c/8/53b14ddbac4d588827291b27c40e3f59eabc24.png[/img] [b]p9 [/b] The numbers 1$, 2, 3, 4, 5, _, 2000$ are written in a row. First, third, fifth, etc. crossed out in order. Of the remaining $1000 $ numbers, the first, third, fifth, etc. are again crossed out. They do this until one number remains. What is this number? [b]p10.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis in $1996$ jumps if he must not get to points with coordinates divisible by $4$ (points $0$, $\pm 4$, $\pm 8$ etc.)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$. *The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$. [i]Proposed by Zack Chroman[/i]

Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that \[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that $$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1.$$ What is the maximum value of $P(a)?$ $\textbf{(A)}\ \frac{7}{12} \qquad\textbf{(B)}\ 2 - \sqrt{2} \qquad\textbf{(C)}\ \frac{1+\sqrt{2}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(E)}\ \frac{5}{8}$

Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$: $xP(x-c) = (x - 2014)P(x)$

Let $ABC$ be a triangle with $A = 90^o, AH$ the altitude, $P,Q$ the feet of the perpendiculars from $H$ to $AB,AC$ respectively. Let $M$ be a variable point on the line $PQ$. The line through $M$ perpendicular to $MH$ meets the lines $AB,AC$ at $R, S$ respectively. i) Prove that circumcircle of $ARS$ always passes the fixed point $H$. ii) Let $M_1$ be another position of $M$ with corresponding points $R_1, S_1$. Prove that the ratio $RR_1/SS_1$ is constant. iii) The point $K$ is symmetric to $H$ with respect to $M$. The line through $K$ perpendicular to the line $PQ$ meets the line $RS$ at $D$. Prove that$ \angle BHR = \angle DHR, \angle DHS = \angle CHS$.

The equation $P(x) = 0$, where $P(x) = x^2+bx+c$, has a single root, and the equation $P(P(P(x))) = 0$ has exactly three different roots. Solve the equation $P(P(P(x))) = 0.$

You have a collection of $\$20.21$, consisting of pennies, nickels, and quarters. To reduce the collection’s worth to $k$ cents, you simultaneously replace all pennies with quarters and all quarters with pennies (all coins are replaced one time). What is the minimum possible $k$? $\textbf{(A) }105\qquad\textbf{(B) }120\qquad\textbf{(C) }125\qquad\textbf{(D) }505\qquad\textbf{(E) }101$

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.

Let $ABC$ be an acute triangle, $k$ its circumcirle and $m$ a line such that $m\cap k=\emptyset, m\parallel BC.$ Denote $D$ the intersection of $m$ and ray $AB.$ a) Let $X$ be an inner point of the arc $BC$ not containing $A$ and denote $Y$ the intersection of lines $m,CX.$ Show that $A,D,X,Y$ are concyclic and name this circle $\kappa$. b) Determine relative position of $\kappa$ and $m$ in case when $C,D,X$ are collinear.

Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \] are all powers of $5$ [i]Proposed by Tejaswi Navilarekallu[/i]

The sequence of real numbers $a_1,a_2,\dots$ is defined as follows: $a_1=56$ and $a_{n+1}=a_n-\frac{1}{a_n}$ for $n\ge 1$. Show that there is an integer $1\leq{k}\leq2002$ such that $a_k<0$.

The diagram below shows a large circle. Six congruent medium-sized circles are each internally tangent to the large circle and tangent to two neighboring medium-sized circles. Three congruent small circles are mutually tangent to one another and are each tangent to two medium-sized circles as shown. The ratio of the area of the large circle to the area of one of the small circles can be written as $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/a/4/fcffd7ee6e8d3da0641525e7a987d13ce05496.png[/img]

A triangle ABC is inscribed in a circle with center $ O$ and radius $ R$. If the inradii of the triangles $ OBC, OCA, OAB$ are $ r_{1}, r_{2}, r_{3}$ , respectively, prove that $ \frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\geq\frac{4\sqrt{3}+6}{R}.$

Consider quadrilateral $ABCD$ with $|DC| > |AD|$. Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$. Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $ \angle ABC$. Let us call $X = L_1 \cap L_2$. Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$ [img]https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png[/img]

Define sequence $(a_{n})_{n=1}^{\infty}$ by $a_1=a_2=a_3=1$ and $a_{n+3}=a_{n+1}+a_{n}$ for all $n \geq 1$. Also, define sequence $(b_{n})_{n=1}^{\infty}$ by $b_1=b_2=b_3=b_4=b_5=1$ and $b_{n+5}=b_{n+4}+b_{n}$ for all $n \geq 1$. Prove that $\exists N \in \mathbb{N}$ such that $a_n = b_{n+1} + b_{n-8}$ for all $n \geq N$.

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Among the numbers $ 11n \plus{} 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?