Two triangles $ABC,ABD$ (with the common side $c=AB$) are given in space. Triangle $ABC$ is right with hypotenuse $AB$, $ABD$ is equilateral. Denote $\varphi$ the dihedral angle between planes $ABC,ABD$. 1) Determine the length of $CD$ in terms of $a=BC,b=CA,c$ and $\varphi$. 2) Determine all values of $\varphi$ such that the tetrahedron $ABCD$ has four sides of the same length.

Let $m,n,a_1,a_2,\dots,a_n$ be positive integers and $r$ be a real number. Prove that the equation \[\lfloor a_1x\rfloor+\lfloor a_2x\rfloor+\cdots+\lfloor a_nx\rfloor=sx+r\] has exactly $ms$ solutions in $x$, where $s=a_1+a_2+\cdots+a_n+\frac1m$. [i]Linus Tang[/i]

Let $m,n$ be natural numbers such that \\ $\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$\\ Find the maximum possible value of $m+n$.

Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter so that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of the circle is $\tfrac{17}{128}$. There are relatively prime positive integers m and n so that the ratio of $DE$ to $AB$ is $\tfrac{m}{n}.$ Find $m + n$.

In an arithmetic sequence, the difference of consecutive terms in constant. We consider sequences of integers in which the difference of consecutive terms equals the sum of the differences of all preceding consecutive terms. Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?

[b]p1.[/b] Find the largest k such that the equation $x^2 - 2x + k = 0$ has at least one real root. [b]p2.[/b] Indiana Jones needs to cross a flimsy rope bridge over a mile long gorge. It is so dark that it is impossible to cross the bridge without a flashlight. Furthermore, the bridge is so weak that it can only support the weight of two people. The party has only one flashlight, which has a weak beam so whenever two people cross, they are constrained to walk together, at the speed of the slower person. Indiana Jones can cross the bridge in $5$ minutes. His girlfriend can cross in $10$ minutes. His father needs $20$ minutes, and his father’s side kick needs $25$ minutes. They need to get everyone across safely in on hour to escape the bay guys. Can they do it? [b]p3.[/b] There are ten big bags with coins. Nine of them contain fare coins weighing $10$ g. each, and one contains counterfeit coins weighing $9$ g. each. By one weighing on a digital scale find the bag with counterfeit coins. [b]p4.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 5x + 5$. [b]p5.[/b] (a) In the $x - y$ plane, analytically determine the length of the path $P \to A \to C \to B \to P$ around the circle $(x - 6)^2 + (y - 8)^2 = 25$ from the point $P(12, 16)$ to itself. [img]https://cdn.artofproblemsolving.com/attachments/f/b/24888b5b478fa6576a54d0424ce3d3c6be2855.png[/img] (b) Determine coordinates of the points $A$ and $B$. [b]p6.[/b] (a) Let $ABCD$ be a convex quadrilateral (it means that diagonals are inside the quadrilateral). Prove that $$Area\,\, (ABCD) \le \frac{|AB| \cdot |AD| + |BC| \cdot |CD|}{2}$$ (b) Let $ABCD$ be an arbitrary quadrilateral (not necessary convex). Prove the same inequality as in part (a). (c) For an arbitrary quadrilateral $ABCD$ prove that $Area\,\, (ABCD) \le \frac{|AB| \cdot |CD| + |BC| \cdot |AD|}{2}$ PS. You should use hide for answers.

Two player are playing in an $100 \times 100$ grid. Initially the whole board is black. On $A$'s move, he selects $4 \times 4$ subgrid and color it white. On $B$'s move, he selects a $3 \times 3$ subgrid and colors it black. $A$ wants to make the whole board white. Can he do it? [i]Proposed by S M A Nahian[/i]

Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \[ \{ n, n+1, n+2, \dots, n+32 \} \] so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)

Let $I$ and $H$ be the incenter and orthocenter of an acute triangle $ABC$. $M$ is the midpoint of arc $AC$ of circumcircle of triangle $ABC$ which does not contain point $B$. If $MI=MH$, find the measure of angle $\angle ABC$. [I]Proposed by F. Bakharev[/i]

Let $n$ be a positive integer. In a village, $n$ boys and $n$ girls are living. For the yearly ball, $n$ dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that $n$ dancing couples can be formed in such a way that every girl is paired with a boy who is on her list. Show that it is possible to form $n$ dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.

What is $ ( \minus{} 1)^1 \plus{} ( \minus{} 1)^2 \plus{} \cdots \plus{} ( \minus{} 1)^{2006}$? $ \textbf{(A) } \minus{} 2006 \qquad \textbf{(B) } \minus{} 1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 2006$

All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.

A parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ is reflected about the $ x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $ y \equal{} f(x)$ and $ y \equal{} g(x)$, respectively. Which of the following describes the graph of $ y \equal{} (f \plus{} g)(x)$? $ \textbf{(A)}\ \text{a parabola tangent to the }x\text{ \minus{} axis}$ $ \textbf{(B)}\ \text{a parabola not tangent to the }x\text{ \minus{} axis} \qquad \textbf{(C)}\ \text{a horizontal line}$ $ \textbf{(D)}\ \text{a non \minus{} horizontal line} \qquad \textbf{(E)}\ \text{the graph of a cubic function}$

As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(real x, real y) { pair P = (x,y); dot(P,linewidth(3)); return P; } int big = 30, small = 14; filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5)); draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare); label("$A$",D2(0,big),NW); label("$B$",D2(0,0),SW); label("$C$",D2(big,0),SW); label("$D$",D2(big,big),N); label("$E$",D2(big+small,0),SE); label("$F$",D2(big+small,small),NE); label("$G$",D2(big,small),NE); [/asy]

If $r > s >0$ and $a > b > c$, prove that \[a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.\]

Initially, there are $14$ numbers written on the board - zeros and ones. Every minute, Anton chooses half of the numbers on the board and adds $1$ to each of them, while Mykhailo multiplies all the other numbers by $8$. At some point (possibly initially), all the numbers on the board become equal. How many ones could have been on the board initially? [i]Proposed by Oleksii Masalitin[/i]

If $r$ is real number sampled at random with uniform probability, find the probability that $r$ is [i]strictly [/i] closer to a multiple of $58$ than it is to a multiple of $37$.

For $a\in \mathbb{Z}$ define \[ n_a=101a-100\cdot 2^a \] Show that, for $0\le a,b,c,d\le 99$ \[ n_a+n_b\equiv n_c+n_d\pmod{10100}\implies \{a,b\}=\{c,d\} \]

Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively. These two circles intersect each other at points $A$ and $B$. Line $O_1B$ intersects $\omega_2$ for the second time at point $C$, and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$. Also $Y$ is the second intersection point of $BD$ and $\omega_2$. Prove that $CX = DY$ . Proposed by Alireza Dadgarnia

Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$ Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$

If the radius of a circle is increased by $ 1$ unit, the ratio of the new circumference to the new diameter is: $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ \frac{2 \pi \plus{} 1}{2} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{2 \pi \minus{} 1}{2} \qquad \textbf{(E)}\ \pi \minus{} 2$

Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.

Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$ [i]Proposed by Alexander Ivanov[/i]

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is $ \textbf{(A)}\ 100 \qquad\textbf{(B)}\ 125 \qquad\textbf{(C)}\ 123 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ \text{none of these} $

Let $ABC$ be a triangle with $BC = a$, $CA = b$, and $AB = c$. The $A$-excircle is tangent to $\overline{BC}$ at $A_1$; points $B_1$ and $C_1$ are similarly defined. Determine the number of ways to select positive integers $a$, $b$, $c$ such that [list] [*] the numbers $-a+b+c$, $a-b+c$, and $a+b-c$ are even integers at most 100, and [*] the circle through the midpoints of $\overline{AA_1}$, $\overline{BB_1}$, and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$. [/list]