In the phrase given at the end of the condition of the problem, it is necessary to put a number (numeral) in place of the ellipsis, written in verbal form and in the required case, so that the statement formulated in it is true. Here is this phrase: “The number of letters in this phrase is...”

The point $D$ is the midpoint of the side $[AB]$ of the triangle $ABC$ . The points $E$ and $F$ belong to $[AC]$ and $[BC]$ respectively. Prove that the area of triangle $DEF$ area does not exceed the sum of the areas of triangles $ADE$ and $BDF$.

A square has coordinates at $(0, 0)$, $(4, 0)$, $(0, 4)$, and $(4, 4)$. Rohith is interested in circles of radius $ r$ centered at the point $(1, 2)$. There is a range of radii $a < r < b$ where Rohith’s circle intersects the square at exactly $6$ points, where $a$ and $b$ are positive real numbers. Then $b - a$ can be written in the form $m +\sqrt{n}$, where $m$ and $n$ are integers. Compute $m + n$.

A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square, The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b\sqrt{2}+\pi)$, where $a$ and $b$ are positive integers. What is $a+b$? [asy] //Diagram by Samrocksnature draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); fill((2,0)--(0,2)--(0,0)--cycle, black); fill((6,0)--(8,0)--(8,2)--cycle, black); fill((8,6)--(8,8)--(6,8)--cycle, black); fill((0,6)--(2,8)--(0,8)--cycle, black); fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); filldraw(circle((2.6,3.31),0.47),gray); [/asy] $\textbf{(A) }64 \qquad \textbf{(B) }66 \qquad \textbf{(C) }68 \qquad \textbf{(D) }70 \qquad \textbf{(E) }72$

Find the minimum possible value of \[\left\lfloor \dfrac{a+b}{c}\right\rfloor+2 \left\lfloor \dfrac{b+c}{a}\right\rfloor+ \left\lfloor \dfrac{c+a}{b}\right\rfloor\] where $a,b,c$ are the sidelengths of a triangle. [i]Proposed by Nathan Ramesh

Solve in $R$ equation $[x] \cdot \{x\} = 2001 x$, where$ [ .]$ and $\{ .\}$ represent respectively the floor and the integer functions.

The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?

Let $ABC$ be an arbitrary scalene triangle. Define $\sum$ to be the set of all circles $y$ that have the following properties: [b](i)[/b] $y$ meets each side of $ABC$ in two (possibly coincident) points; [b](ii)[/b] if the points of intersection of $y$ with the sides of the triangle are labeled by $P, Q, R, S, T , U$, with the points occurring on the sides in orders $\mathcal B(B,P,Q,C), \mathcal B(C, R, S,A), \mathcal B(A, T,U,B)$, then the following relations of parallelism hold: $TS \parallel BC; PU\parallel CA; RQ\parallel AB$. (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if $A$ lies on the circle $y$, then $T$ , $S$ both coincide with $A$ and the relation $TS \parallel BC$ holds vacuously.) [i](a)[/i] Under what circumstances is $\sum$ nonempty? [i](b)[/i] Assuming that Σ is nonempty, show how to construct the locus of centers of the circles in the set $\sum$. [i](c)[/i] Given that the set $\sum$has just one element, deduce the size of the largest angle of $ABC.$ [i](d)[/i] Show how to construct the circles in $\sum$ that have, respectively, the largest and the smallest radii.

Let $P$ be a nonempty collection of subsets of $\{1,\dots,n\}$ such that: (i) if $S,S'\in P,$ then $S\cup S'\in P$ and $S\cap S'\in P,$ and (ii) if $S\in P$ and $S\ne\emptyset,$ then there is a subset $T\subset S$ such that $T\in P$ and $T$ contains exactly one fewer element than $S.$ Suppose that $f:P\to\mathbb{R}$ is a function such that $f(\emptyset)=0$ and \[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\] Must there exist real numbers $f_1,\dots,f_n$ such that \[f(S)=\sum_{i\in S}f_i\] for every $S\in P?$

In an isosceles triangle $ABC$ with $BC=AC$ and $\angle B<60^\circ$, $I$ is the incenter and $O$ the circumcenter. The circle with center $E$ that passes through $A,O$ and $I$ intersects the circumcircle of $\triangle ABC$ again at point $D$. Prove that the lines $DE$ and $CO$ intersect on the circumcircle of $ABC$.

In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$. Prove that $X,Y,Z$ are collinear. [i]Proposed by Hooman Fattahi[/i]

$\forall k\in N \,\,\, \exists n(k) \in N, a(k):0<a(k)<1 [(1+\sqrt2)^{2k+1}=n(k)+a(k)]$ Prove: $(n(k) + a(k))a(k) = 1$, for all $k \in N$, and calculate $\lim_{k \to \infty }a(k)$

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

Find the maximum value of $ \dfrac{x}{y} $ if $ x $ and $ y $ are real numbers such that $ x^2 + y^2 - 8x - 6y + 20 = 0 $.

Determine all integer solutions of the equation $x^n+y^n=1994$ where $n\geq 2$

Let $a, b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$.

Prove that for any three distinct positive real numbers $a, b$ and $c$: $$\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}> 8abc.$$

Consider $111$ distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least $1998$ segments formed by these points with length $\leq \sqrt{3}$

If $ \frac{m}{n}\equal{}\frac{4}{3}$ and $ \frac{r}{t}\equal{}\frac{9}{14}$, the value of $ \frac{3mr\minus{}nt}{4nt\minus{}7mr}$ is: $ \textbf{(A)}\ \minus{}5 \frac{1}{2} \qquad \textbf{(B)}\ \minus{}\frac{11}{14} \qquad \textbf{(C)}\ \minus{}1\frac{1}{4} \qquad \textbf{(D)}\ \frac{11}{14} \qquad \textbf{(E)}\ \minus{}\frac{2}{3}$

Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.

$n\geq 4$ is an integer number. For any permutation $x_1,x_2,\cdots,x_n$ of the numbers $1,2 \cdots,n$ we write the number $$ x_1+2x_2+\cdots+nx_n $$ on the board. Compute the number of total distinct numbers written on the board.

Let $ABCD$ be a parallelogram and let $E$, $F$, $G$, and $H$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. If $BH \cap AC = I$, $BD \cap EC = J$, $AC \cap DF = K$, and $AG \cap BD = L$, prove that the quadrilateral $IJKL$ is a parallelogram.

(a) An infinite sheet is divided into squares by two sets of parallel lines. Two players play the following game: the first player chooses a square and colours it red, the second player chooses a non-coloured square and colours it blue, the first player chooses a non-coloured square and colours it red, the second player chooses a non-coloured square and colours it blue, and so on. The goal of the first player is to colour four squares whose vertices form a square with sides parallel to the lines of the two parallel sets. The goal of the second player is to prevent him. Can the first player win? (b) What is the answer to this question if the second player is permitted to colour two squares at once? (DG Azov) PS. (a) for Juniors, (a),(b) for Seniors

A convex quadrilateral $ABCD$ is constructed out of metal rods with negligible thickness. The side lengths are $AB=BC=CD=5$ and $DA=3$. The figure is then deformed, with the angles between consecutive rods allowed to change but the rods themselves staying the same length. The resulting figure is a convex polygon for which $\angle{ABC}$ is as large as possible. What is the area of this figure? $\text{(A) }6\qquad\text{(B) }8\qquad\text{(C) }9\qquad\text{(D) }10\qquad\text{(E) }12$

In the expression $ \frac{x \plus{} 1}{x \minus{} 1}$ each $ x$ is replaced by $ \frac{x \plus{} 1}{x \minus{} 1}$. The resulting expression, evaluated for $ x \equal{} \frac{1}{2}$, equals: $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ \minus{}3\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \minus{}1\qquad \textbf{(E)}\ \text{none of these}$