There exist integers $a$ and $b$ such that $(1 +\sqrt2)^{12}= a + b\sqrt2$. Compute the remainder when $ab$ is divided by $13$.

The convex set $F$ does not cover a semi-circle of radius $R$. Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ? What if $F$ is not convex? ( N . B . Vasiliev , A. G . Samosvat)

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.

Paul is in the desert and has a pile of gypsum crystals. No matter how he divides the pile into two nonempty piles, at least one of the resulting piles has a number of crystals that, when written in base $10,$ has a sum of digits at least $7.$ Given that Paul’s initial pile has at least two crystals, compute the smallest possible number of crystals in the initial pile.

Cerena, Faith, Edna, and Veronica each have a cube. Aarnő knows that the side lengths of each of their cubes are distinct integers greater than $1$, and he is trying to guess their exact values. Each girl fully paints the surface of her cube in Carolina blue before splitting the entire cube into $1\times1\times1$ cubes. Then, [list=disc] [*] Cerena reveals how many of her $1\times1\times1$ cubes have exactly $0$ blue faces. [*] Faith reveals how many of her $1\times1\times1$ cubes have exactly $1$ blue faces. [*] Edna reveals how many of her $1\times1\times1$ cubes have exactly $2$ blue faces. [*] Veronica reveals how many of her $1\times1\times1$ cubes have exactly $3$ blue faces. [/list] Whose side lengths can Aarnő deduce from these statements? [i]Jason Lee[/i]

Consider the following graph of position vs. time, which represents the motion of a certain particle in the given potential. [asy] import roundedpath; size(300); picture pic; // Rectangle draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle); label(pic,"0",(0,0),S); label(pic,"2",(4,0),S); label(pic,"4",(8,0),S); label(pic,"6",(12,0),S); label(pic,"8",(16,0),S); label(pic,"10",(20,0),S); label(pic,"-15",(0,2),W); label(pic,"-10",(0,4),W); label(pic,"-5",(0,6),W); label(pic,"0",(0,8),W); label(pic,"5",(0,10),W); label(pic,"10",(0,12),W); label(pic,"15",(0,14),W); label(pic,rotate(90)*"x (m)",(-2,7),W); label(pic,"t (s)",(11,-2),S); // Tick Marks draw(pic,(4,0)--(4,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(12,0)--(12,0.3)); draw(pic,(16,0)--(16,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(4,15)--(4,14.7)); draw(pic,(8,15)--(8,14.7)); draw(pic,(12,15)--(12,14.7)); draw(pic,(16,15)--(16,14.7)); draw(pic,(20,15)--(20,14.7)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(0,12)--(0.3,12)); draw(pic,(0,14)--(0.3,14)); draw(pic,(20,2)--(19.7,2)); draw(pic,(20,4)--(19.7,4)); draw(pic,(20,6)--(19.7,6)); draw(pic,(20,8)--(19.7,8)); draw(pic,(20,10)--(19.7,10)); draw(pic,(20,12)--(19.7,12)); draw(pic,(20,14)--(19.7,14)); // Path add(pic); path A=(0.102, 6.163)-- (0.192, 6.358)-- (0.369, 6.500)-- (0.526, 6.642)-- (0.643, 6.712)-- (0.820, 6.830)-- (0.938, 6.901)-- (1.075, 7.043)-- (1.193, 7.185)-- (1.369, 7.256)-- (1.506, 7.374)-- (1.644, 7.445)-- (1.840, 7.515)-- (1.958, 7.586)-- (2.134, 7.657)-- (2.291, 7.752)-- (2.468, 7.846)-- (2.625, 7.846)-- (2.899, 7.893)-- (3.095, 8.035)-- (3.350, 8.035)-- (3.586, 8.106)-- (3.860, 8.106)-- (4.135, 8.106)-- (4.371, 8.035)-- (4.606, 8.035)-- (4.881, 8.012)-- (5.155, 7.917)-- (5.391, 7.823)-- (5.665, 7.728)-- (5.960, 7.563)-- (6.175, 7.468)-- (6.332, 7.374)-- (6.528, 7.232)-- (6.725, 7.161)-- (6.882, 6.996)-- (7.117, 6.854)-- (7.333, 6.712)-- (7.509, 6.523)-- (7.666, 6.358)-- (7.902, 6.146)-- (8.098, 5.980)-- (8.274, 5.791)-- (8.451, 5.649)-- (8.647, 5.484)-- (8.882, 5.248)-- (9.196, 5.059)-- (9.392, 4.894)-- (9.628, 4.752)-- (9.824, 4.634)-- (10.118, 4.516)-- (10.452, 4.350)-- (10.785, 4.232)-- (11.001, 4.185)-- (11.315, 4.138)-- (11.648, 4.114)-- (12.002, 4.114)-- (12.257, 4.091)-- (12.610, 4.067)-- (12.825, 4.161)-- (13.081, 4.185)-- (13.316, 4.279)-- (13.492, 4.327)-- (13.689, 4.445)-- (13.826, 4.516)-- (14.022, 4.587)-- (14.159, 4.705)-- (14.316, 4.823)-- (14.532, 4.964)-- (14.669, 5.059)-- (14.866, 5.177)-- (15.062, 5.248)-- (15.278, 5.461)-- (15.474, 5.697)-- (15.650, 5.838)-- (15.847, 6.004)-- (16.043, 6.169)-- (16.258, 6.334)-- (16.415, 6.523)-- (16.592, 6.736)-- (16.788, 6.830)-- (17.063, 7.067)-- (17.357, 7.232)-- (17.573, 7.397)-- (17.808, 7.515)-- (18.063, 7.634)-- (18.358, 7.704)-- (18.573, 7.870)-- (18.887, 7.941)-- (19.142, 8.012)-- (19.358, 8.035)-- (19.574, 8.082)-- (19.770, 8.130); draw(shift(1.8*up)*roundedpath(A,0.09),linewidth(1.5)); [/asy] What is the total energy of the particle? (A) $\text{-5 J}$ (B) $\text{0 J}$ (C) $\text{5 J}$ (D) $\text{10 J}$ (E) $\text{15 J}$

How many $\textit{odd}$ four-digit integers have the property that their digits, read left to right, are in strictly decreasing order?

Harold tosses a nickel four times. The probability that he gets at least as many heads as tails is $\text{(A)}\ \frac{5}{16} \qquad \text{(B)}\ \frac{3}{8} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{5}{8} \qquad \text{(E)}\ \frac{11}{16}$

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is [i]fibonatic[/i] when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not [i]fibonatic[/i] integers.

[b]E[/b]milia wishes to create a basic solution with 7% hydroxide (OH) ions. She has three solutions of different bases available: 10% rubidium hydroxide (Rb(OH)), 8% cesium hydroxide (Cs(OH)), and 5% francium hydroxide (Fr(OH)). (The Rb(OH) solution has both 10% Rb ions and 10% OH ions, and similar for the other solutions.) Since francium is highly radioactive, its concentration in the final solution should not exceed 2%. What is the highest possible concentration of rubidium in her solution?

A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$. [i]S. Tokarev[/i]

Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$. [i]Proposed by Bojan Bašić, Serbia[/i]

Geetha wants to cut a cube of size $4 \times 4\times 4$ into $64$ unit cubes (of size $1\times 1\times 1$). Every cut must be straight, and parallel to a face of the big cube. What is the minimum number of cuts that Geetha needs? Note: After every cut, she can rearrange the pieces before cutting again. At every cut, she can cut more than one pieces as long as the pieces are on a straight line.

$a, b$ and $c$ are positive integers and \[\frac{a\sqrt{3} + b}{b\sqrt{3} + c}\] is a rational number. Show that \[\frac{a^2 + b^2 + c^2}{a + b + c}\] is an integer.

Bob is practicing addition in base $2.$ Each time he adds two numbers in base $2,$ he counts the number of carries. For example, when summing the numbers $1001$ and $1011$ in base $2,$ \[\begin{array}{ccccc} \overset{1}{}&& \overset {1}{}&\overset {1}{} \\ 0&1&0&0&1\\0&1&0&1&1 \\ \hline 1&0&1&0&0 \end{array}\] there are three carries (shown on the top row). Suppose that Bob starts with the number $0,$ and adds $111~($i.e. $7$ in base $2)$ to it one hundred times to obtain the number $1010111100~($i.e. $700$ in base $2).$ How many carries occur (in total) in these one hundred calculations? \[\mathrm a. ~ 280\qquad \mathrm b.~289\qquad \mathrm c. ~291 \qquad \mathrm d. ~294 \qquad \mathrm e. ~297\]

Let $ ABCD$ be a parallelogram with no angle equal to $ 60^{\textrm{o}}$. Find all pairs of points $ E, F$, in the plane of $ ABCD$, such that triangles $ AEB$ and $ BFC$ are isosceles, of basis $ AB$, respectively $ BC$, and triangle $ DEF$ is equilateral. [i]Valentin Vornicu[/i]

Let $n$ be a positive integer divisible by $4$. We consider the permutations $(a_1, a_2,...,a_n)$ of $(1,2,..., n)$ having the following property: for each j we have $a_i + j = n + 1$ where $i = a_j$ . Prove that there are exactly $\frac{ (\frac12 n)!}{(\frac14 n)!}$ such permutations.

For $p_0,\dots,p_d\in\mathbb{R}^d$, let \[ S(p_0,\dots,p_d)=\left\{ \alpha_0p_0+\dots+\alpha_dp_d : \alpha_i\le 1, \sum_{i=0}^d \alpha_i =1 \right\}. \] Let $\pi$ be an arbitrary probability distribution on $\mathbb{R}^d$, and choose $p_0,\dots,p_d$ independently with distribution $\pi$. Prove that the expectation of $\pi(S(p_0,\dots,p_d))$ is at least $1/(d+2)$.

A triangle $ABC$ is given on the plane, such that all its vertices have integer coordinates. Does there necessarily exist a straight line which intersects the straight lines $AB,$ $BC,$ and $AC$ at three distinct points with integer coordinates?

Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions: \[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\] Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$

What is the greatest possible value of c such that $x^2+5x+c=0$ has at least one real solution?

A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$. She also has a box with dimensions $10\times11\times14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out. Find all possible values of the total number of bricks that she can pack.

Let $m=\frac{-1+\sqrt{17}}{2}$. Let the polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is given, where $n$ is a positive integer, the coefficients $a_0,a_1,a_2,...,a_n$ are positive integers and $P(m) =2018$ . Prove that the sum $a_0+a_1+a_2+...+a_n$ is divisible by $2$ .

Let P be the common point of 3 internal bisectors of a given ABC: The line passing through P and perpendicular to CP intersects AC and BC at M and N, respectively. If AP = 3cm, BP = 4cm, compute the value of $\frac{AM}{BN}$ ?