For what values of the ratio $a/b$ is the limaçon $r = a - b \cos \theta$ a convex curve? $(a > b > 0)$

A large bag of coins contains pennies, dimes, and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is $ \textbf{(A)}\ \$306 \qquad \textbf{(B)}\ \$333 \qquad \textbf{(C)}\ \$342 \qquad \textbf{(D)}\ \$348 \qquad \textbf{(E)}\ \$360$

$ x$ and $ y$ are two distinct positive integers. What is the minimum positive integer value of $ (x \plus{} y^2)(x^2 \minus{} y)/(xy)$ ? $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17$

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Ten cars are moving at the road. There are some cities at the road. Each car is moving with some constant speed through cities and with some different constant speed outside the cities (different cars may move with different speed). There are 2011 points at the road. Cars don't overtake at the points. Prove that there are 2 points such that cars pass through these points in the same order. [i]S. Berlov[/i]

Circles $O_1$ and $O_2$ interest at two points $ B$ and $ C,$ and $ BC$ is the diameter of circle $O_1.$ Construct a tangent line of circle $O_1$ at $ C$ and intersecting circle $O_2$ at another point $ A.$ We join $ AB$ to intersect circle $O_1$ at point $ E,$ then join $ CE$ and extend it to intersect circle $O_2$ at point $ F.$ Assume $ H$ is an arbitrary point on line segment $ AF.$ We join $ HE$ and extend it to intersect circle $O_1$ at point $ G,$ and then join $ BG$ and extend it to intersect the extend line of $ AC$ at point $ D.$ Prove that \[ \frac{AH}{HF} = \frac{AC}{CD}.\]

Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as $$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate: $$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$

In isosceles triangle $ABC$ ($AB = BC$) one draws the angle bisector $CD$. The perpendicular to $CD$ through the center of the circumcircle of $ABC$ intersects $BC$ at $E$. The parallel to $CD$ through $E$ meets $AB$ at $F$. Show that $BE$ = $FD$. [i]M. Sonkin[/i]

We say that a quadrilateral $Q$ is [i]tangential [/i] if a circle can be inscribed into it, i.e. there exists a circle $C$ that does not meet the vertices of $Q$, such that it meets each edge at exactly one point. Let $N$ be the number of ways to choose four distinct integers out of $\{1, . . . , 24\}$ so that they form the side lengths of a tangential quadrilateral. Find the largest prime factor of $N$.

A lazy student used the approximation $\pi=\frac{22} 7$ to calculate the circumference of a given circle. If his answer was 6, what was the radius of the circle?

$H$ is the orthocenter of a non-isosceles acute triangle $\triangle ABC$. $M$ is the midpoint of $BC$ and $BB_1, CC_1$ are two altitudes of $\triangle ABC$. $N$ is the midpoint of $B_1C_1$. Prove that $AH$ is tangent to the circumcircle of $\triangle MNH$.

Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$.

Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$, find (a)max$S$ and (b) min$S$.

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$? $\textbf{(A) } 9 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 13 \qquad\textbf{(E) } 15 $

Let $f$ be a polynomial with real coefficients and suppose $f$ has no nonnegative real root. Prove that there exists a polynomial $h$ with real coefficients such that the coefficients of $fh$ are nonnegative.

Let ${AB}$ be a diameter of a circle ${\omega}$ and center ${O}$ , ${OC}$ a radius of ${\omega}$ perpendicular to $AB$,${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points ${M,O,P,N}$ are cocyclic. (Albania)

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

Prove that (a) if the natural number $n$ can be represented as $n =4k+1$ (where $k$ is an integer), then there exist $n$ odd positive integers whose sum is equal to their product, (b) if $n$ cannot be represented in this form then such a set does not exist. (M. Kontsevich)

Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that: $ (i)$ $ a\plus{}c\equal{}d;$ $ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$ $ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$. Determine $ n$.

Let $ABC$ be a triangle with $AH$ altitude. The point $K$ is chosen on the segment $AH$ as follows such that $AH =3KH$. Let $O$ be the center of the circle circumscribed around by triangle $ABC, M$ and $N$ be the midpoints of $AC$ and AB respectively. Lines $KO$ and $MN$ intersect at the point $Z$, a perpendicular to $OK$ passing through point $Z$ intersects lines $AC$ and $AB$ at points $X$ and $Y$ respectively. Prove that $\angle XKY =\angle CKB$.

Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that \[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.

Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length $16,$ find the radius of the large circle.

The figure below consists of several unit squares, $M$ of which are white and $N$ of which are green. Compute $100M+N$. [asy] size(4cm); int N = 4; path square; for (int x=-N; x<=N; ++x) { for (int y=-N+abs(x); y<=N-abs(x); ++y) { square = rotate(9)*((x+0.5,y+0.5)--(x+0.5,y-0.5)--(x-0.5,y-0.5)--(x-0.5,y+0.5)--cycle); if ((x+y) % 2 == 0) { filldraw(square, green, black); } else { filldraw(square, white, black); } } } [/asy] [i]Proposed by Evan Chen[/i]

It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$. From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are same as first $k$ elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element $1$.