Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.

Let a complete oriented graph on $n$ points be given. Show that the vertices can be enumerated as $v_1 , v_2 ,\ldots, v_n$ such that $v_1 \rightarrow v_2 \rightarrow \cdots \rightarrow v_n.$

The circle $\omega$ inscribed in an isosceles triangle $ABC$ ($AC = BC$) touches the side $BC$ at point $D$ .On the extensions of the segment $AB$ beyond points $A$ and $B$, respectively mark the points $K$ and $L$ so that $AK = BL$, The lines $KD$ and $LD$ intersect the circle $\omega$ for second time at points $G$ and $H$, respectively. Prove that point $A$ belongs to the line $GH$.

A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ and circumscribed to a circle with center $I$. Its diagonals meet at $P$. Prove that points $O, I$ and $P$ lie on a line.

Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$

Let $a, b$ and $c$ be real numbers such that $ab + bc + ca> a + b + c> 0$. Show then that $a+b+c>3$

Given any triangle $ABC$ and any positive integer $n$, we say that $n$ is a [i]decomposable[/i] number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$. Determine the positive integers that are decomposable numbers for every triangle.

The repeating decimal $2.0151515\ldots$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.

Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.

For positive intenger $n$, define $f(n)$: the smallest positive intenger that does not divide $n$. Consider sequence $(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3)$. For example, $a_3=a_2+1=2,a_4=a_3+1=3$. [b](a)[/b] Prove that there exists a positive intenger $C$, for any positive intenger $n$, $a_n\leq C$. [b](b)[/b] Are there positive intengers $M$ and $T$, satisfying that for any positive intenger $n\geq M$, $a_n=a_{n+T}$.

Two transformations are said to [i]commute[/i] if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane: - A translation $2$ units to the right - A $90^\circ$- rotation counterclockwise about the origin. - A reflection across the $x$-axis, and - A dilation centered at the origin with scale factor $2$. Of the $6$ pairs of distinct transformations from this list, how many commute? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5 \qquad $

Consider a circle $k$ and a point $K$ in the plane. For any two distinct points $P$ and $Q$ on $k$, denote by $k'$ the circle through $P,Q$ and $K$. The tangent to $k'$ at $K$ meets the line $PQ$ at point $M$. Describe the locus of the points $M$ when $P$ and $Q$ assume all possible positions.

Let $a, b$ be positive real numbers, and let $x, y$ be complex numbers such that $|x| = a$ and $|y| = b$. Find the minimal and maximal value of \[\left|\frac{x + y}{1 + x\overline{y}}\right|\]

In a convex pentagon $ABCDE$, the points $M$, $N$, $P$, $Q$, $R$ are the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EA$, respectively. If the segments $AP$, $BQ$, $CR$ and $DM$ pass through a single point, prove that $EN$ contains that point as well. [i]Yugoslavia[/i]

Let $ ABCD$ be a rhombus with $ AC\equal{}16$ and $ BD\equal{}30$. Let $ N$ be a point on $ \overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \overline{AC}$ and $ \overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ PQ$? [asy]unitsize(2.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C; pair Np=waypoint(B--A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(Np--Q); draw(Np--P); label("$D$",D,SW); label("$C$",C,SE); label("$B$",B,NE); label("$A$",A,NW); label("$N$",Np,N); label("$P$",P,SW); label("$Q$",Q,SSE); draw(rightanglemark(Np,P,C,2)); draw(rightanglemark(Np,Q,D,2));[/asy]$ \textbf{(A)}\ 6.5 \qquad \textbf{(B)}\ 6.75 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 7.25 \qquad \textbf{(E)}\ 7.5$

The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.

Suppose that 13 cards numbered $1, 2, 3, \dots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes? [asy] size(11cm); draw((0,0)--(2,0)--(2,3)--(0,3)--cycle); label("7", (1,1.5)); draw((3,0)--(5,0)--(5,3)--(3,3)--cycle); label("11", (4,1.5)); draw((6,0)--(8,0)--(8,3)--(6,3)--cycle); label("8", (7,1.5)); draw((9,0)--(11,0)--(11,3)--(9,3)--cycle); label("6", (10,1.5)); draw((12,0)--(14,0)--(14,3)--(12,3)--cycle); label("4", (13,1.5)); draw((15,0)--(17,0)--(17,3)--(15,3)--cycle); label("5", (16,1.5)); draw((18,0)--(20,0)--(20,3)--(18,3)--cycle); label("9", (19,1.5)); draw((21,0)--(23,0)--(23,3)--(21,3)--cycle); label("12", (22,1.5)); draw((24,0)--(26,0)--(26,3)--(24,3)--cycle); label("1", (25,1.5)); draw((27,0)--(29,0)--(29,3)--(27,3)--cycle); label("13", (28,1.5)); draw((30,0)--(32,0)--(32,3)--(30,3)--cycle); label("10", (31,1.5)); draw((33,0)--(35,0)--(35,3)--(33,3)--cycle); label("2", (34,1.5)); draw((36,0)--(38,0)--(38,3)--(36,3)--cycle); label("3", (37,1.5)); [/asy] $\textbf{(A) }4082\qquad\textbf{(B) }4095\qquad\textbf{(C) }4096\qquad\textbf{(D) }8178\qquad\textbf{(E) }8191$

Kamil wrote on a board an expression consisting of alternating addition and subtraction of natural numbers from $1$ to $100$: \[1-2+3-4+5-6+\ldots-98+99-100.\] Then, Kamil erased one of the plus or minus signs and replaced it with an equals sign, obtaining a true equality. Which number preceded the erased sign? Find all possibilities and justify your answer.

Diameter $AB$ of a circle has length a 2-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.

Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$? $ \textbf{(A) }5 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }15 \qquad $

Let $n$ and $k$ be positive integers and $G$ be a complete graph on $n$ vertices. Each edge of $G$ is colored one of $k$ colors such that every triangle consists of either three edges of the same color or three edges of three different colors. Furthermore, there exist two different-colored edges. Prove that $n\le(k-1)^2$. [i]Linus Tang[/i]

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$. [i]Proposed by Evan Chen[/i]

Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions: a) not a triple of successive members is chosen, b) at least one of every triple of successive members is chosen, c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.

[u]Set 3[/u] [b]3.1[/b] Annie has $24$ letter tiles in a bag; $8$ C’s, $8$ M’s, and $8$ W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw? [b]3.2[/b] Let $T$ be the answer from the previous problem. Charlotte is initially standing at $(0, 0)$ in the coordinate plane. She takes $T$ steps, each of which moves her by $1$ unit in either the $+x$, $-x$, $+y$, or $-y$ direction (e.g. her first step takes her to $(1, 0)$, $(1, 0)$, $(0, 1)$ or $(0, -1)$). After the T steps, how many possibilities are there for Charlotte’s location? [b]3.3[/b] Let $T$ be the answer from the previous problem, and let $S$ be the sum of the digits of $T$. Francesca has an unfair coin with an unknown probability $p$ of landing heads on a given flip. If she flips the coin $S$ times, the probability she gets exactly one head is equal to the probability she gets exactly two heads. Compute the probability $p$. PS. You should use hide for answers.