Let \(ABC\) be a triangle and \(D\) be the mid-point of \(BC\). Suppose the angle bisector of \(\angle ADC\) is tangent to the circumcircle of triangle \(ABD\) at \(D\). Prove that \(\angle A=90^{\circ}\).

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.

The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$.

If the integer $k$ is added to each of the numbers $36$, $300$, and $596$, one obtains the squares of three consecutive terms of an arithmetic series. Find $k$.

$\triangle ABC$ has side lengths $AB=20$, $BC=15$, and $CA=7$. Let the altitudes of $\triangle ABC$ be $AD$, $BE$, and $CF$. What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$?

Let $ABC$ be a triangle inscribed in the circle $(O)$. Let $I$ be the center of the circle inscribed in the triangle and $D$ the point of contact of the circle inscribed with the side $BC$. Let $M$ be the second intersection point of the bisector $AI$ with the circle $(O)$ and let $P$ be the point where the line $DM$ intersects the circle $(O)$ . Show that $PA \perp PI$.

In the diagram, two circles, each with center D, have radii of $1$ and $2$. The total area of the shaded region is $\frac{5}{12}$ of the area of the larger circle. How many degrees are in the measure of $\angle ADC$? [asy] int angle = 100; path A = arc(0, 1, 0, angle); path B = arc(0, 1, angle, 360); path C = arc(0, 2, 0, angle); path D = arc(0, 2, angle, 360); filldraw(C -- origin -- cycle, gray); filldraw(D -- origin -- cycle, white); filldraw(A -- origin -- cycle, white); filldraw(B -- origin -- cycle, gray); label("$D$", origin, NE); label("$C$", (2, 0), E); label("$A$", (2, 0) * dir(angle), N); [/asy] $\mathrm{(A) \ } 100 \qquad \mathrm{(B) \ } 105 \qquad \mathrm {(C) \ } 110 \qquad \mathrm{(D) \ } 115 \qquad \mathrm{(E) \ } 120$

Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$. Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible. (It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)

Is it possible to locate in a rectangle of $5$ cm by $ 8$ cm, $51$ circles of diameter $ 1$ cm, so that they don't overlap? Could it be possible for more than $40$ circles ?

The triangle with side lengths $3, 5$, and $k$ has area $6$ for two distinct values of $k$: $x$ and $y$. Compute $|x^2 -y^2|$.

Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red. Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$ [I]Netherlands[/i]

Let $T$ be a triangle with side lengths 3, 4, and 5. If $P$ is a point in or on $T$, what is the greatest possible sum of the distances from $P$ to each of the three sides of $T$?

Let $x,y,z$ represent the side lengths of any triangle, and $s=\dfrac{x+y+z}{2}$ and the area of this triangle be $\sqrt{s}$ square units. Prove that $$s\Big(\frac{1}{x(s-x)^2}+\frac{1}{y(s-y)^2}+\frac{1}{z(s-z)^ 2} \Big)\ge \frac{1}{2} \Big(\frac{1}{s-x}+\frac{1}{s-y}+\frac{1}{s-z}\Big)$$ [i](Zhuge Liang)[/i]

Let $k$ and $s$ be odd positive integers such that \[\sqrt{3k-2}-1 \le s \le \sqrt{4k}.\] Show that there are nonnegative integers $t$, $u$, $v$, and $w$ such that \[k=t^{2}+u^{2}+v^{2}+w^{2}, \;\; \text{and}\;\; s=t+u+v+w.\]

Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.

[asy]size(100); draw((0,0)--(5,0),MidArrow); draw((5,0)--(10,0),MidArrow); draw((5,5sqrt(3))--(2.5,2.5sqrt(3)),MidArrow); draw((2.5,2.5sqrt(3))--(0,0),MidArrow); draw((5,5sqrt(3))--(7.5,2.5sqrt(3)),MidArrow); draw((7.5,2.5sqrt(3))--(10,0),MidArrow); draw((7.5,2.5sqrt(3))--(2.5,2.5sqrt(3)),MidArrow); draw((7.5,2.5sqrt(3))--(5,0),MidArrow); draw((2.5,2.5sqrt(3))--(5,0),MidArrow); label("D",(0,0),SW); label("C",(5,0),S); label("N",(10,0),SE); label("A",(2.5,2.5sqrt(3)),W); label("B",(7.5,2.5sqrt(3)),E); label("M",(5,5sqrt(3)),N);[/asy] Using only the paths and the directions shown, how many different routes are there from $ M$ to $ N$? \[ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6 \]

The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits $1$ through $9$ in such a way that the sum of the numbers on every three consecutive vertices is a multiple of $3$. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.

Real numbers $a, b, c, d, e$, have the property $$|a - b| = 2|b -c| = 3|c - d| = 4|d- e| = 5|e - a|.$$ Prove they are all equal.

We define $A, B, C$ as a [i]partition[/i] of $\mathbb{N}$ if $A,B,C$ satisfies the following. (i) $A, B, C \not= \phi$ (ii) $A \cap B = B \cap C = C \cap A = \phi$ (iii) $A \cup B \cup C = \mathbb{N}$. Prove that the partition of $\mathbb{N}$ satisfying the following does not exist. (i) $\forall$ $a \in A, b \in B$, we have $a+b+2008 \in C$ (ii) $\forall$ $b \in B, c \in C$, we have $b+c+2008 \in A$ (iii) $\forall$ $c \in C, a \in A$, we have $c+a+2008 \in B$

If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\] then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.

We say that a positive integer $n$ is circular if it is possible to place the numbers $1, 2, \cdots , n$ in a circumference so that there are no three adjacent numbers whose sum is a multiple of 3. a) Show that 9 is not circular b) Show that any integer greater than 9 is circular.

Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.

The sequence $ (a_n)$ satisfies $ a_0 \equal{} 0$ and $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$ for $ n\ge0$. Find the greatest integer less than or equal to $ a_{10}$.

For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0 < x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of \[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j + 1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.

In every cell of a board $9 \times 9$ is written an integer. For any $k$ numbers in the same row (column), their sum is also in the same row (column). Find the smallest possible number of zeroes in the board for $a)$ $k=5;$ $b)$ $k=8.$