Let $ABCD$ be a square pyramid of height $\frac{1}{2}$ with square base $ABCD$ of side length $AB=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $ABCD$ is the center of square $ABCD$). The faces $ADE$ and $CDE$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$). Find $\tan \alpha$.

Let $ABC$ be an acute triangle with $AB<AC < BC$, inscirbed in circle $\Gamma_1$, with center $O$. Circle $\Gamma_2$, with center point $A$ and radius $AC$ intersects $BC$ at point $D$ and the circle $\Gamma_1$ at point $E$. Line $AD$ intersects circle $\Gamma_1$ at point $F$. The circumscribed circle $\Gamma_3$ of triangle $DEF$, intersects $BC$ at point $G$. Prove that: a) Point $B$ is the center of circle $\Gamma_3$ b) Circumscribed circle of triangle $CEG$ is tangent to $AC$.

A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of $4$. The square of the length of the minor axis of the ellipse can be written in the form $a + b\sqrt{c}$ where $a, b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find the sum $a + b + c$.

The vertices of a $50\text{-gon}$ divide a circumference into $50$ arcs, whose lengths are $1,2,\ldots 50$ in some order. It is known that any two opposite arcs (corresponding to opposite sides) differ by $25$. Prove that the polygon has two parallel sides.

A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. Prove that if $np < k$ then there is a vertex in $B$ that can be joined to all vertices of $A$.

A combination lock has a $3$ number combination, with each number an integer between $0$ and $39$ inclusive. Call the numbers $n_1$, $n_2$, and $n_3$. If you know that $n_1$ and $n_3$ leave the same remainder when divided by $4$, and $n_2$ and $n_1 + 2$ leave the same remainder when divided by $4$, how many possible combinations are there?

Find all integers $n \ge 3$ with the following property: it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$. Poland

How many triangles are in this figure? (Some triangles may overlap other triangles.) [asy] draw((0,0)--(42,0)--(14,21)--cycle); draw((14,21)--(18,0)--(30,9));[/asy] $ \text{(A)}\ 9\qquad\text{(B)}\ 8\qquad\text{(C)}\ 7\qquad\text{(D)}\ 6\qquad\text{(E)}\ 5 $

p1. A set of cards contains $100$ cards, each of which is written with a number from $1$ up to $100$. On each of the two sides of the card the same number is written, side one is red and the other is green. First of all Leny arranges all the cards with red writing face up. Then Leny did the following three steps: I. Turn over all cards whose numbers are divisible by $2$ II. Turn over all the cards whose numbers are divisible by $3$ III. Turning over all the cards whose numbers are divisible by $5$, but didn't turn over all cards whose numbers are divisible by $5$ and $2$. Find the number of Leny cards now numbered in red and face up, p2. Find the area of ​​three intersecting semicircles as shown in the following image. [img]https://cdn.artofproblemsolving.com/attachments/f/b/470c4d2b84435843975a0664fad5fee4a088d5.png[/img] p3. It is known that $x+\frac{1}{x}=7$ . Determine the value of $A$ so that $\frac{Ax}{x^4+x^2+1}=\frac56$. p4. There are $13$ different gifts that will all be distributed to Ami, Ima, Mai,and Mia. If Ami gets at least $4$ gifts, Ima and Mai respectively got at least $3$ gifts, and Mia got at least $2$ gifts, how many possible gift arrangements are there? p5. A natural number is called a [i]quaprimal [/i] number if it satisfies all four following conditions: i. Does not contain zeros. ii. The digits compiling the number are different. iii. The first number and the last number are prime numbers or squares of an integer. iv. Each pair of consecutive numbers forms a prime number or square of an integer. For example, we check the number $971643$. (i) $971643$ does not contain zeros. (ii) The digits who compile $971643$ are different. (iii) One first number and one last number of $971643$, namely $9$ and $3$ is a prime number or a square of an integer. (iv) Each pair of consecutive numbers, namely $97, 71, 16, 64$, and $43$ form prime number or square of an integer. So $971643$ is a quadratic number. Find the largest $6$-digit quaprimal number. Find the smallest $6$-digit quaprimal number. Which digit is never contained in any arbitrary quaprimal number? Explain.

Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ such that \[f(xy + x + y) + f \left( \frac1x \right) f\left( \frac1y \right) = 1\] for every $x$, $y\in \mathbb{R}_+$. [i]Proposed by Li4 and Untro368.[/i]

Prove that for every natural $n$ $$\frac{1}{3} + \frac{2}{3\cdot 5} + \frac{3}{3 \cdot 5 \cdot 7} + ...+ \frac{n}{3 \cdot 5 \cdot 7 \cdot ...\cdot (2n+1)} < \frac{1}{2}.$$

A circle is drawn on the plane. How to use only a ruler to draw a perpendicular from a given point outside the circle to a given line passing through the center of this circle?

Find all positive integers $n$ such that \[2n = \varphi(n)^{\frac{2}{3}}(\varphi(n)^{\frac{2}{3}}+1)\]

Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$. $(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$. $(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.

Given points $A, B, C$ on a line, equilateral triangles $ABC_1$ and $BCA_1$ constructed on segments $AB$ and $BC$, and midpoints $M$ and $N$ of $AA_1$ and $CC_1$, respectively. Prove that $\vartriangle BMN$ is equilateral. (We assume that $B$ lies between $A$ and $C$, and points $A_1$ and $C_1$ lie on the same side of line $AB$)

Let $ ABC $ be a triangle in which $ O,I, $ are the circumcenter, respectively, incenter. The mediators of $ IA,IB,IC, $ form a triangle $ A_1B_1C_1. $ Show that $ \overrightarrow{OI}=\overrightarrow{OA_1} +\overrightarrow{OA_2} +\overrightarrow{OA_3} . $

There are two equal non-intersecting circles on a plane. Two lines were drawn. Each of the lines intersects the circles at four points so that the three segments formed on each of the lines are equal (segments with ends at adjacent points of intersection are considered). For one line these segments have length $a$, for the other they have length $b$ ($a < b$). Find the radius of the circles.

[b]p1.[/b] How many primes $p < 100$ satisfy $p = a^2 + b^2$ for some positive integers $a$ and $b$? [b]p2. [/b] For $a < b < c$, there exists exactly one Pythagorean triple such that $a + b + c = 2000$. Find $a + c - b$. [b]p3.[/b] Five points lie on the surface of a sphere of radius $ 1$ such that the distance between any two points is at least $\sqrt2$. Find the maximum volume enclosed by these five points. [b]p4.[/b] $ABCDEF$ is a convex hexagon with $AB = BC = CD = DE = EF = FA = 5$ and $AC = CE = EA = 6$. Find the area of $ABCDEF$. [b]p5.[/b] Joe and Wanda are playing a game of chance. Each player rolls a fair $11$-sided die, whose sides are labeled with numbers $1, 2, ... , 11$. Let the result of the Joe’s roll be $X$, and the result of Wanda’s roll be $Y$ . Joe wins if $XY$ has remainder $ 1$ when divided by $11$, and Wanda wins otherwise. What is the probability that Joe wins? [b]p6.[/b] Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number $n$ is divisible by $3$, he replaces $n$ with $\frac{n}{3} + 2$, and otherwise he replaces $n$ with $\lfloor 3 \log_3 n \rfloor$. If he starts with the number $3^{2011}$, what number will he have after $2011$ steps? Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p7.[/b] Define a sequence an of positive real numbers with a$_1 = 1$, and $$a_{n+1} =\frac{4a^2_n - 1}{-2 + \frac{4a^2_n -1}{-2+ \frac{4a^2_n -1}{-2+...}}}.$$ What is $a_{2011}$? [b]p8.[/b] A set $S$ of positive integers is called good if for any $x, y \in S$ either $x = y$ or $|x - y| \ge 3$. How many subsets of $\{1, 2, 3, ..., 13\}$ are good? Include the empty set in your count. [b]p9.[/b] Find all pairs of positive integers $(a, b)$ with $a \le b$ such that $10 \cdot lcm \, (a, b) = a^2 + b^2$. Note that $lcm \,(m, n)$ denotes the least common multiple of $m$ and $n$. [b]p10.[/b] For a natural number $n$, $g(n)$ denotes the largest odd divisor of $n$. Find $$g(1) + g(2) + g(3) + ... + g(2^{2011})$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What algebraic relationship holds between $ A $, $ B $, and $ C $ if $$ctg A + \frac{\cos B}{\sin A \cos C} = ctg B + \frac{\cos A}{\sin B \cos C}.$$

Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?

Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.) $\textbf{(A) } 12\qquad\textbf{(B) } 15\qquad\textbf{(C) }17 \qquad\textbf{(D) }20 \qquad\textbf{(E) }25$

Compute the sum of all positive integers whose digits form either a strictly increasing or strictly decreasing sequence.

Segments connect vertices $A, B, C$ of $\vartriangle ABC$ with respective points $A_1, B_1, C_1$ on the opposite sides of the triangle. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ do not belong to one straight line.

Let \( n \) be a positive integer. A function \( f : \{0, 1, \dots, n\} \to \{0, 1, \dots, n\} \) is called \( n \)-Bolivian if it satisfies the following conditions: • \( f(0) = 0 \); • \( f(t) \in \{ t-1, f(t-1), f(f(t-1)), \dots \} \) for all \( t = 1, 2, \dots, n \). For example, if \( n = 3 \), then the function defined by \( f(0) = f(1) = 0 \), \( f(2) = f(3) = 1 \) is 3-Bolivian, but the function defined by \( f(0) = f(1) = f(2) = 0 \), \( f(3) = 1 \) is not 3-Bolivian. For a fixed positive integer \( n \), Gollum selects an \( n \)-Bolivian function. Smeagol, knowing that \( f \) is \( n \)-Bolivian, tries to figure out which function was chosen by asking questions of the type: \[ \text{How many integers } a \text{ are there such that } f(a) = b? \] given a \( b \) of his choice. Show that if Gollum always answers correctly, Smeagol can determine \( f \) and find the minimum number of questions he needs to ask, considering all possible choices of \( f \).

Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?