We have 19 triminos (2 x 2 squares without one unit square) and infinite amount of 2 x 2 squares. Find the greatest odd number $n$ for which a square $n$ x $n$ can be covered with the given figures.

Jerry has a four-sided die, a six-sided die, and an eight-sided die. Each die is numbered starting at one. Jerry rolls the three dice simultaneously. What is the probability that they all show different numbers? $\mathrm{(A) \ } \frac{35}{48} \qquad \mathrm{(B) \ } \frac{35}{64} \qquad \mathrm {(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{5}{12} \qquad \mathrm{(E) \ } \frac{5}{8}$

[b]a)[/b] Show that $ \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, $ for all non-negative reals $ x $ and integers $ n\ge 2. $ [b]b)[/b] If $ x,y,z\in (0,\infty ) , $ then prove the inequality $$ \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16 $$

Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB$, $\angle OBA$, $\angle OBC$, $\angle OCB$, $\angle OAC$, $\angle OCA$ are acute.

Let $ABC$ be a triangle. Point $D$ lies on segment $BC$ such that $\angle BAD = \angle DAC$. Point $X$ lies on the opposite side of line $BC$ as $A$ and satisfies $XB=XD$ and $\angle BXD = \angle ACB$. The point $Y$ is defined similarly. Prove that the lines $XY$ and $AD$ are perpendicular.

$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.

What is the perimeter of trapezoid $ ABCD$? [asy]defaultpen(linewidth(0.8));size(3inch, 1.5inch); pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0); draw(a--b--c--d--cycle); draw(b--e); draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e); label("30", (9,12), W); label("50", (43,24), N); label("25", (71.5, 12), E); label("24", (18, 12), E); label("$A$", a, SW); label("$B$", b, N); label("$C$", c, N); label("$D$", d, SE); label("$E$", e, S);[/asy] $ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 188\qquad\textbf{(C)}\ 196\qquad\textbf{(D)}\ 200\qquad\textbf{(E)}\ 204 $

The altitudes of an acute triangle $ABC$ with $AB<AC$ intersect at a point $H$, and $O$ is the center of the circumcircle $\Omega$. The segment $OH$ intersects the circumcircle of $BHC$ at a point $X$, different from $O$ and $H$. The circumcircle of $AOX$ intersects the smaller arc $AB$ of $\Omega$ at point $Y$. Prove that the line $XY$ bisects the segment $BC$. [i]Proposed by A. Tereshin[/i]

$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$. Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$, and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$. Prove that $L, J, A, Q,$ and $K$ all lie on a circle.

Let $P(z)$ be a polynomial with real coefficients whose roots are covered by a disk of radius R. Prove that for any real number $k$, the roots of the polynomial $nP(z)-kP'(z)$ can be covered by a disk of radius $R+|k|$, where $n$ is the degree of $P(z)$, and $P'(z)$ is the derivative of $P(z)$. can anyone help me? It would also be extremely helpful if anyone could tell me where they've seen this type of problems.............Has it appeared in any mathematics competitions? Or are there any similar questions for me to attempt? Thanks in advance!

Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$.

Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.) Evaluate $ \lim_{n\to\infty}d_n.$

Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying : For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.

$A$ and $B$ are $2\times 2$ real valued matrices satisfying $$\det A = \det B = 1,\quad \text{tr}(A)>2,\quad \text{tr}(B)>2,\quad \text{tr}(ABA^{-1}B^{-1}) = 2$$ Prove that $A$ and $B$ have a common eigenvector.

Let $x_1,\ldots,x_n$ be a sequence with each term equal to $0$ or $1$. Form a triangle as follows: its first row is $x_1,\ldots,x_n$ and if a row if $a_1, a_2, \ldots, a_m$, then the next row is $a_1 + a_2, a_2 + a_3, \ldots, a_{m-1} + a_m$ where the addition is performed modulo $2$ (so $1+1=0$). For example, starting from $1$, $0$, $1$, $0$, the second row is $1$, $1$, $1$, the third one is $0$, $0$ and the fourth one is $0$. A sequence is called good it is the same as the sequence formed by taking the last element of each row, starting from the last row (so in the above example, the sequence is $1010$ and the corresponding sequence from last terms is $0010$ and they are not equal in this case). How many possibilities are there for the sequence formed by taking the first element of each row, starting from the last row, which arise from a good sequence?

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

Initially $A$ selects a graph with \( 2221 \) vertices such that each vertex is incident to at least one edge. Then $B$ deletes some of the edges (possibly none) from the chosen graph. Finally, $A$ pays $B$ one lev for each vertex that is incident to an odd number of edges. What is the maximum amount that $B$ can guarantee to earn?

The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata? [asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W)); path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy] $\textbf{(A) } \dfrac{4\pi}{5} \qquad\textbf{(B) } \dfrac{9\pi}{8} \qquad\textbf{(C) } \dfrac{4\pi}{3} \qquad\textbf{(D) } \dfrac{7\pi}{5} \qquad\textbf{(E) } \dfrac{3\pi}{2} $

Let $\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx$. (a) Prove that $s_n \leq \dfrac 2n$ for all odd $n$. (b) Find all the limit points of the sequence $s_1, s_2, s_3, \dots$. [i]Proposed by Cristi Calin[/i]

Find all positive integers n such that : $-2^{0}+2^{1}-2^{2}+2^{3}-2^{4}+...-(-2)^{n}=4^{0}+4^{1}+4^{2}+...+4^{2010}$

$(x_1, y_1), (x_2, y_2), (x_3, y_3)$ lie on a straight line and on the curve $y^2 = x^3$. Show that $\frac{x_1}{y_1} + \frac{x_2}{y_2}+\frac{x_3}{y_3} = 0$.

In triangle $ABC$, in which $AB = BC$, on side $AB$ is selected point $D$, and the ciscumcircles of triangles $ADC$ and $BDC$ , $S1$ and $S2$ respectively. The tangent drawn to $S_1$ at point $D$ intersects $S_2$ for second time at point $M$. Prove that $BM \parallel AC$.

Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \perp AC$ and $EF \parallel BC$. Prove that $FB = FD$. [i]Milan Haiman[/i]

$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that : $ |T| \leq \frac {4}{9}n + \log_2 n + 2$