Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$. (I.Voronovich)

A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

In the City of Islands there are $2021$ islands connected by bridges. For any given pair of islands $A$ and $B$, one can go from island $A$ to island $B$ using the bridges. Moreover, for any four islands $A_1, A_2, A_3$ and $A_4$: if there is a bridge from $A_i$ to $A_{i+1}$ for each $i \in \left \{ 1,2,3 \right \}$, then there is a bridge between $A_{j}$ and $A_{k}$ for some $j,k \in \left \{ 1,2,3,4 \right \}$ with $|j-k|=2$. Show that there is at least one island which is connected to any other island by a bridge.

Find $$\lim_{x\rightarrow0}\frac{\sin(x)-x}{x\cos(x)-x}.$$

Let $N = 2^{\left(2^2\right)}$ and $x$ be a real number such that $N^{\left(N^N\right)} = 2^{(2^x)}$. Find $x$.

In the diagram, $ABDF$ is a trapezoid with $AF$ parallel to $BD$ and $AB$ perpendicular to $BD.$ The circle with center $B$ and radius $AB$ meets $BD$ at $C$ and is tangent to $DF$ at $E.$ Suppose that $x$ is equal to the area of the region inside quadrilateral $ABEF$ but outside the circle, that y is equal to the area of the region inside $\triangle EBD$ but outside the circle, and that $\alpha = \angle EBC.$ Prove that there is exactly one measure $\alpha,$ with $0^\circ \leq \alpha \leq 90^\circ,$ for which $x = y$ and that this value of $\frac 12 < \sin \alpha < \frac{1}{\sqrt 2}.$ [asy] import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen fftttt = rgb(1,0.2,0.2); draw(circle((6.04,2.8),1.78),qqttff); draw((6.02,4.58)--(6.04,2.8),fftttt); draw((6.02,4.58)--(6.98,4.56),fftttt); draw((6.04,2.8)--(8.13,2.88),fftttt); draw((6.98,4.56)--(8.13,2.88),fftttt); dot((6.04,2.8),ds); label("$B$", (5.74,2.46), NE*lsf); dot((6.02,4.58),ds); label("$A$", (5.88,4.7), NE*lsf); dot((6.98,4.56),ds); label("$F$", (7.06,4.6), NE*lsf); dot((7.39,3.96),ds); label("$E$", (7.6,3.88), NE*lsf); dot((8.13,2.88),ds); label("$D$", (8.34,2.56), NE*lsf); dot((7.82,2.86),ds); label("$C$", (7.5,2.46), NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy]

A chord $AB$ is drawn in a certain circle. The smaller of the two arcs $AB$ corresponds to a central angle of $120^o$. A tangent $p$ to this arc is drawn. Two circles with radii $R$ and $r$ are constructed, touching this smaller arc $AB$ and straight lines $AB$ and $p$. Find the radius of the original circle.

Find all real numbers $x$ such that $(x-1)^2(x-4)^2<(x-2)^2$.

Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .

The ordered pair of four-digit numbers $(2025, 3136)$ has the property that each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the first number. Find, with proof, all ordered pairs of five-digit numbers and ordered pairs of six-digit numbers with the same property: each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the first number.

There are $K$ boys placed around a circle. Each of them has an even number of sweets. At a command each boy gives half of his sweets to the boy on his right. If, after that, any boy has an odd number of sweets, someone outside the circle gives him one more sweet to make the number even. This procedure can be repeated indefinitely. Prove that there will be a time at which all boys will have the same number of sweets. (A Andjans, Riga)

Consider the real numbers $a_1,a_2,...,a_{2n}$ whose sum is equal to $0$. Prove that among pairs $(a_i,a_j) , i<j$ where $ i,j \in \{1,2,...,2n\} $ .there are at least $2n-1$ pairs with the property that $a_i+a_j\ge 0$.

Let $A{}$ be a point in the Cartesian plane. At each step, Ann tells Bob a number $0< a\leqslant 1$ and he then moves $A{}$ in one of the four cardinal directions, at his choice, by a distance of $a{}$. This process cotinues as long as Ann wishes. Amongst every $100$ consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than $100$ from the initial position of $A{}$. Can Ann achieve her goal?

If $\sin 2x \sin 3x = \cos 2x \cos 3x$, then one value for $x$ is A. $18^\circ$ B. $30^\circ$ C. $36^\circ$ D. $45^\circ$ E. $60^\circ$

The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\phi$ between the cone’s generating line and its base.

At the base of the triangular pyramid $ABCD$ lies a regular triangle $ABC$ such that $AD = BC$. All plane angles at vertex $B$ are equal to each other. What might these angles be equal to?

On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$, such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$. [i]Nicolae Bourbacut[/i]

The graphs of $ x^2 \plus{} y^2 \equal{} 4 \plus{} 12x \plus{} 6y$ and $ x^2 \plus{} y^2 \equal{} k \plus{} 4x \plus{} 12y$ intersect when $ k$ satisfies $ a \leq k \leq b$, and for no other values of $ k$. Find $ b \minus{} a$. $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 68\qquad \textbf{(C)}\ 104\qquad \textbf{(D)}\ 140\qquad \textbf{(E)}\ 144$

Is there a function $f$ from the positive integers to themselves such that $$f(a)f(b) \geq f(ab)f(1)$$ with equality [b]if and only if[/b] $(a-1)(b-1)(a-b)=0$?

Suppose that $p$ and $q$ are distinct primes and $S$ is a subset of $\{1, 2, ..., p-1\}$. Let $N(S)$ denote the number of ordered $q$-tuples $(x_1,x_2,...,x_q)$ with $x_i \in S$, $1 \le i \le q$, such that $x_1+x_2+...+x_q \cong 0 (mod p)$.

Find the number of positive integers which divide $10^{999}$ but not $10^{998}$.

p1. Circle $M$ is the incircle of ABC, while circle $N$ is the incircle of $ACD$. Circles $M$ and $N$ are tangent at point $E$. If side length $AD = x$ cm, $AB = y$ cm, $BC = z$ cm, find the length of side $DC$ (in terms of $x, y$, and $z$). [img]https://cdn.artofproblemsolving.com/attachments/d/5/66ddc8a27e20e5a3b27ab24ff1eba3abee49a6.png[/img] p2. The address of the house on Jalan Bahagia will be numbered with the following rules: $\bullet$ One side of the road is numbered with consecutive even numbers starting from number $2$. $\bullet$ The opposite side is numbered with an odd number starting from number $3$. $\bullet$ In a row of even numbered houses, there is some land vacant house that has not been built. $\bullet$ The first house numbered $2$ has a neighbor next door. When the RT management ordered the numbers of the house, it is known that the cost of making each digit is $12.000$ Rp. For that, the total cost to be incurred is $1.020.000$ Rp. It is also known that the cost of all even-sided house numbers is $132.000$ Rp. cheaper than the odd side. When the land is empty later a house has been built, the number of houses on the even and odd sides is the same. Determine the number of houses that are now on Jalan Bahagia . p3. Given the following problem: Each element in the set $A = \{10, 11, 12,...,2008\}$ multiplied by each element in the set $B = \{21, 22, 23,...,99\}$. The results are then added together to give value of $X$. Determine the value of $X$. Someone answers the question by multiplying $2016991$ with $4740$. How can you explain that how does that person make sense? p4. Let $P$ be the set of all positive integers between $0$ and $2008$ which can be expressed as the sum of two or more consecutive positive integers . (For example: $11 = 5 + 6$, $90 = 29 + 30 + 31$, $100 = 18 + 19 +20 + 21 + 22$. So $11, 90, 100$ are some members of $P$.) Find the sum of of all members of $P$. p5. A four-digit number will be formed from the numbers at $0, 1, 2, 3, 4, 5$ provided that the numbers in the number are not repeated, and the number formed is a multiple of $3$. What is the probability that the number formed has a value less than $3000$?

Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.

Let $x, y, z$ be real numbers such that $$\displaystyle{\begin{cases} x^2+y^2+z^2+(x+y+z)^2=9 \\ xyz \leq \frac{15}{32} \end{cases}} $$ Find the maximum possible value of $x.$

From given positive numbers, the following infinite sequence is defined: $a_1$ is the sum of all original numbers, $a_2$ is the sum of the squares of all original numbers, $a_3$ is the sum of the cubes of all original numbers, and so on ($a_k$ is the sum of the $k$-th powers of all original numbers). a) Can it happen that $a_1 > a_2 > a_3 > a_4 > a_5$ and $a_5 < a_6 < a_7 < \ldots$? (4 points) b) Can it happen that $a_1 < a_2 < a_3 < a_4 < a_5$ and $a_5 > a_6 > a_7 > \ldots$? (4 points) [i](Alexey Tolpygo)[/i]