Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$

Alphonse and Beryl play a game starting with a blank blackboard. Alphonse goes first and the two players alternate turns. On Alphonse's first turn, he writes the integer $10^{2011}$ on the blackboard. On each subsequent turn, each player can do exactly one of the following two things: [b](i)[/b] replace any number $x$ that is currently on the blackboard with two integers a and b greater than $1$ such that $x = ab,$ or [b](ii)[/b] erase one or two copies of a number $y$ that appears at least twice on the blackboard. Thus, there may be many numbers on the board at any time. The first player who cannot do either of these things loses. Determine which player has a winning strategy and explain the strategy.

Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$?

The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\ \[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\] for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$.

Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

Given is a natural number $n\geqslant 3$. Find the smallest $k{}$ for which the following statement is true: for any $n{}$-gon and any two points inside it there is a broken line with $k{}$ segments connecting these points, lying entirely inside the $n{}$-gon. [i]Proposed by L. Emelyanov[/i]

A closed broken self-intersecting line is drawn in the plane. Each of the links of this line is intersected exactly once and no three links intersect at the same point. Further, there are no self-intersections at the vertices and no two links have a common segment. Can it happen that every point of self-intersection divides both links in halves?

Define that $x*y=ax+by+cxy$ for all real numbers $x,y$, where $a,b,c$ are uncertain. It is known that $1*2=3,2*3=4$. If there exists a real number $d$, for any real number $x$, $x*d=x$, then $d=$________.

We are given a semicircle $k$ with center $O$ and diameter $AB$. Let $C$ be a point on $k$ such that $CO \bot AB$. The bisector of $\angle ABC$ intersects $k$ at point $D$. Let $E$ be a point on $AB$ such that $DE \bot AB$ and let $F$ be the midpoint of $CB$. Prove that the quadrilateral $EFCD$ is cyclic.

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

Set $T\subset\{1,2,\dots,n\}^3$ has the property that for any two triplets $(a,b,c)$ and $(x,y,z)$ in $T$, we have $a<b<c$, and also, we know that at most one of the equalities $a=x$, $b=y$, $c=z$ holds. Maximize $|T|$.

In a village (where only dwarfs live) there are $k$ streets, and there are $k(n-1)+1$ clubs each containing $n$ dwarfs. A dwarf can be in more than one clubs, and two dwarfs know each other if they live in the same street or they are in the same club (there is a club they are both in). Prove that is it possible to choose $n$ different dwarfs from $n$ different clubs (one dwarf from each club), such that the $n$ dwarfs know each other!

Let $f(x)=x+3x^{\frac 23}, g(x)=x+x^{\frac 13}$. Call a sequence $\{a_i\}_{i\ge 0}$ satisfactory if for all $i\ge 1, a_i\in \{f(a_{i-1}), g(a_{i-1})\}$. Find all pairs of real numbers $(x,y)$ such that there exist satisfactory sequences $(a_i)_{i\ge 0}, (b_i)_{i\ge 0}$ and positive integers $m$ and $n$, such that $a_0 =x$, $b_0 = y$, and $$|a_m-b_n|<1$$

Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$.

If $x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20$ then $x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} = $ $ \textbf{(A)}\ 5.05\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 51.005\qquad\textbf{(D)}\ 61.25\qquad\textbf{(E)}\ 400 $

Prove that for every triangle the following inequality holds: \[\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.\] where $a, b, c$ are lengths of the sides and $S$ is the area of the triangle.

Is it true that there exists a triangle with sides $x, y, z$ so that $x^3+y^3+z^3=(x+y)(y+z)(z+x)$?

Let $S$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form \[0.abcabcabc\ldots=0.\overline{abc},\] where the digits $a$, $b$, and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required?

Consider a point $M$ inside triangle $ABC$ such that triangles $ABM, BCM$ and $CAM$ have equal areas. Prove that $M$ is the intersection point of the medians of triangle $ABC$.

Let be a natural number $ n\ge 2, $ two complex numbers $ p,q, $ and four matrices $ A,B,C,D\in\mathcal{M}_n(\mathbb{C}) $ such that $ A+B=C+D=pI,AB+CD=qI $ and $ ABCD=0. $ Show that $ BCDA=0. $ [i]Marius Cavachi[/i]

Find the formula to express the number of $ n\minus{}$series of letters which contain an even number of vocals (A,I,U,E,O).

In an acute triangle $ABC$, a point $P$ is taken on the $A$-altitude. Lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at points $D$ and $E$, respectively. Tangents drawn from points $D$ and $E$ to the circumcircle of triangle $BPC$ are tangent to it at points $K$ and $L$, respectively, which are in the interior of triangle $ABC$. Line $KD$ intersects the circumcircle of triangle $AKC$ at point $M$ for the second time, and line $LE$ intersects the circumcircle of triangle $ALB$ at point $N$ for the second time. Prove that\[ \frac{KD}{MD}=\frac{LE}{NE} \iff \text{Point P is the orthocenter of triangle ABC}\]

Regular hexagon is divided to equal rhombuses, with sides, parallels to hexagon sides. On the three sides of the hexagon, among which there are no neighbors, is set directions in order of traversing the hexagon against hour hand. Then, on each side of the rhombus, an arrow directed just as the side of the hexagon parallel to this side. Prove that there is not a closed path going along the arrows.