Which of the following is equivalent to $ \displaystyle \sqrt {\frac {x}{1 \minus{} \frac {x \minus{} 1}{x}}}$ when $ x < 0$? $ \textbf{(A) } \minus{} x \qquad \textbf{(B) } x \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \sqrt {\frac x2} \qquad \textbf{(E) } x\sqrt { \minus{} 1}$

Let $ M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace$. Find the smallest $ n \in \mathbb{N}$ such that any $ n$-element subset of $ M$ contains 3 pairwise disjoint 4-element subsets $ S, T, U$ such that [b]I.[/b] For any 2 elements in $ S$, the larger number is a multiple of the smaller number. The same applies for $ T$ and $ U$. [b]II.[/b] For any $ s \in S$ and $ t \in T$, $ (s,t) \equal{} 1$. [b]III.[/b] For any $ s \in S$ and $ u \in U$, $ (s,u) > 1$.

The $42$ points $P_1,P_2,\ldots,P_{42}$ lie on a straight line, in that order, so that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1\leq n\leq41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)

$A$ and $B$ are any two subsets of $\{1, 2,...,n - 1\}$ such that $|A| +|B|> n - 1$. Prove that one can find $a$ in $A$ and $b$ in $B$ such that $a + b = n$.

Let $\Omega$ be a unit circle with diameter $AB$ and center $O$. Let $C$, $D$ be on $\Omega$ and lie on the same side of $AB$ such that $\angle CAB = 50^\circ$ and $\angle DBA = 70^\circ$. Suppose $AD$ intersects $BC$ at $E$. Let the perpendicular from $O$ to $CD$ intersect the perpendicular from $E$ to $AB$ at $F$. Find the length of $OF$. [i]Proposed by Puhua Cheng[/i]

Let $ABC$ be an acute triangle with circumcircle $\Omega$ and $\overline{AB} < \overline{AC}$. The angle bisector of $A$ meets $\Omega$ again at $D$, and the line through $D$, perpendicular to $BC$ meets $\Omega$ again at $E$. The circle centered at $A$, passing through $E$ meets the line $DE$ again at $F$. Let $K$ be the circumcircle of triangle $ADF$. Prove that $AK$ is perpendicular to $BC$.

There are $19$ lines in the plane dividing the plane into exactly $97$ pieces. (a) Prove that among these pieces there is at least one triangle. (b) Show that it is indeed possible to place $19$ lines in the above way.

Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]

A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 95% and the probability of him finishing his chemistry problem set that night is 75%. If the student worked on his problem sets in the the chemistry library, the probability of him finishing his chemistry problem set that night is 90% and the probability of him finishing his math problem set that night is 80%. Since he had no bicycle, he could only work in one of the libraries on February 23rd. He works in the math library with a probability of 60%. Given that he finished both problem sets that night, what is the probability that he worked on the problem sets in the math library?

The apparatus in the diagram consists of a solid cylinder of radius $\text{1}$ cm attached at the center to two disks of radius $\text{2}$ cm. It is placed on a surface where it can roll, but will not slip. A thread is wound around the central cylinder. When the thread is pulled at the angle $\theta = \text{90}^{\circ}$ to the horizontal (directly up), the apparatus rolls to the right. Which below is the largest value of $\theta$ for which it will not roll to the right when pulling on the thread? 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filldraw(roundedpath(E,0.35),lightgray); draw((0,0)--(-11.830, 8.804),EndArrow(size=13)); picture pic; draw(pic,circle((0,0),3)); draw(pic,circle((0,0),7)); draw(pic,(-2.5,-1.66)--(-8,9),EndArrow(size=9)); draw(pic,(-2.5,-1.66)--(-7.5,-1.66),dashed); label(pic,scale(0.75)*"$\theta$",(-4.5,-1.5),N); add(shift(35*right)*shift(3*up)*pic); [/asy] (A) $\theta = \text{15}^{\circ}$ (B) $\theta = \text{30}^{\circ}$ (C) $\theta = \text{45}^{\circ}$ (D) $\theta = \text{60}^{\circ}$ (E) None, the apparatus will always roll to the right

In the accompanying figure $ \overline{CE}$ and $ \overline{DE}$ are equal chords of a circle with center $ O$. Arc $ AB$ is a quarter-circle. Then the ratio of the area of triangle $ CED$ to the area of triangle $ AOB$ is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair C = (-1,0); pair D = (1,0); pair E = (0,1); pair A = dir(-135); pair B = dir(-60); draw(Circle(O,1)); draw(C--E--D--cycle); draw(A--O--B--cycle); label("$A$",A,SW); label("$C$",C,W); label("$E$",E,N); label("$D$",D,NE); label("$B$",B,SE); label("$O$",O,N);[/asy] $ \textbf{(A)}\ \sqrt {2} : 1\qquad \textbf{(B)}\ \sqrt {3} : 1\qquad \textbf{(C)}\ 4 : 1\qquad \textbf{(D)}\ 3 : 1\qquad \textbf{(E)}\ 2 : 1$

Let $AA_1, BB_1, $ and $CC_1$ be the bisectors of a triangle $ABC$. The segments $BB_1$ and $A_1C_1$ meet at point $D$. Let $E$ be the projection of $D$ to $AC$. Points $P$ and $Q$ on sides $AB$ and $BC$ respectively are such that $EP = PD, EQ = QD$. Prove that $\angle PDB_1 = \angle EDQ$.

If $a$, $b$, $c > 0$; $ab + bc + ca = 3$ then: $$4\left(\tan^{-1} 2\right)\left(\tan^{-1}\left(\sqrt[3]{abc}\right)\right) \leq \pi \tan^{-1}\left(1 + \sqrt[3]{abc}\right)$$

Let $n$ $\geq$ $2$ and $k$ $\geq$ $2$ be positive integers. A cat and a mouse are playing [i]Wim[/i], which is a stone removal game. The game starts with $n$ stones and they take turns removing stones, with the cat going first. On each turn they are allowed to remove $1$, $2$, $\dotsb$, or $k$ stones, and the player who cannot remove any stones on their turn loses. \\\\ A raccoon finds Wim very boring and creates [i]Wim 2[/i], which is Wim but with the following additional rule: [i]You cannot remove the same number of stones that your opponent removed on the previous turn[/i]. \\\\Find all values of $k$ such that for every $n$, the cat has a winning strategy in Wim if and only if it has a winning strategy in Wim 2.

In a country, there are some cities linked together by roads. The roads just meet each other inside the cities. In each city, there is a board which showing the shortest length of the road originating in that city and going through all other cities (the way can go through some cities more than one times and is not necessary to turn back to the originated city). Prove that 2 random numbers in the boards can't be greater or lesser than 1.5 times than each other.

Prove that there is an integer $n>10^{2018}$ such that the sum of all primes less than $n$ is relatively prime to $n$. [i](R. Salimov)[/i]

If the number of triangles that $\angle ABC=60^{\circ},AC=12,BC=k$ is exactly one, then the range value of $k$ is $\text{(A)}k=8\sqrt3\qquad\text{(B)}k=0<k\leq12\qquad\text{(C)}k\geq12\qquad\text{(D)}k=8\sqrt3\text { or }0<k\leq12$

Three distinct integers are chosen uniformly at random from the set $$\{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}.$$ Compute the probability that their arithmetic mean is an integer.

The lengths of the sides of triangle $A'B'C'$ are equal to the lengths of the three medians of triangle $ABC.$ Then the ratio $\mathrm{Area} (A'B'C') / \mathrm{Area} (ABC)$ equals $$ \mathrm a. ~ \frac 12\qquad \mathrm b.~\frac 23\qquad \mathrm c. ~\frac34 \qquad \mathrm d. ~\frac56 \qquad \mathrm e. ~\text{Cannot be determined from the information given.} $$

Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.

[b]p1.[/b] Solve the equation $x^2 - x - cos y+1.25 =0$. [b]p2.[/b] Solve the inequality: $\left| \frac{x - 2}{x - 3}\right| \le x$ [b]p3.[/b] Bilbo and Dwalin are seated at a round table of radius $R$. Bilbo places a coin of radius $r$ at the center of the table, then Dwalin places a second coin as near to the table’s center as possible without overlapping the first coin. The process continues with additional coins being placed as near as possible to the center of the table and in contact with as many coins as possible without overlap. The person who places the last coin entirely on the table (no overhang) wins the game. Assume that $R/r$ is an integer. (a) Who wins, Bilbo or Dawalin? Please justify your answer. (b) How many coins are on the table when the game ends? [b]p4.[/b] In the center of a square field is an orc. Four elf guards are on the vertices of that square. The orc can run in the field, the elves only along the sides of the square. Elves run $\$1.5$ times faster than the orc. The orc can kill one elf but cannot fight two of them at the same time. Prove that elves can keep the orc from escaping from the field. [b]p5.[/b] Nine straight roads cross the Mirkwood which is shaped like a square, with an area of $120$ square miles. Each road intersects two opposite sides of the square and divides the Mirkwood into two quadrilaterals of areas $40$ and $80$ square miles. Prove that there exists a point in the Mirkwood which is an intersection of at least three roads. PS. You should use hide for answers.

The sequence $n_1<n_2<\ldots < n_k$ consists of all positive integers $n$ for which in a square $n \times n$ one can mark $10$ cells such that in any square $3 \times 3$ an odd amount of cells are marked. Find $n_{k-2}$.

Prove that, if every three consecutive vertices of a convex $n{}$-gon, $n\geqslant 4$, span a triangle of area at least 1, then the area of the $n{}$-gon is (strictly) greater than $(n\log_2 n)/4-1/2.$ [i]Radu Bumbăcea & Călin Popescu[/i]

Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.

For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by \[ W(n,k) = \begin{cases} n^n & k = 0 \\ W(W(n,k-1), k-1) & k > 0. \end{cases} \] Find the last three digits in the decimal representation of $W(555,2)$.