On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse's total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly? $ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}\ 31\qquad\textbf{(E)}\ 33$

For how many ordered quadruplets $(a,b,c,d)$ of positive integers such that $2\leq a\leq b \leq c$ and $1 \leq d \leq 418$ do we have that $bcd+abd+acd=abc+abcd?$

A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$. (Day 1, 2nd problem authors: Michal Rolínek, Jaroslav Švrček)

Find the number of ways to arrange the letters in $LE X I NGTON$ such that the string $LE X$ does not appear.

A $12$-sided polygon is inscribed in a circle with radius $r$. The polygon has six sides of length $6\sqrt3$ that alternate with six sides of length $2$. Find $r^2$.

Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles

Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows: $$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$ Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.

For any positive reals $x$, $y$, $z$ with $xyz + xy + yz + zx = 4$, prove that $$\sqrt{\frac{xy+x+y}{z}}+\sqrt{\frac{yz+y+z}{x}}+\sqrt{\frac{zx+z+x}{y}}\geq 3\sqrt{\frac{3(x+2)(y+2)(z+2)}{(2x + 1)(2y + 1)(2z + 1). }}$$

Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ that satisfy $f(x+y)-f(x-y)=2y(3x^2+y^2)$ for all $x,y{\in}R$ ______________________________________ Azerbaijan Land of the Fire :lol:

Let $f$ be a real-valued function such that $4f(x)+xf\left(\tfrac1x\right)=x+1$ for every positive real number $x$. Find $f(2014)$.

[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? [b]b.)[/b] For which $n>2$ is there exactly one set having this property?

Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?

Let $m, n$ integers such that: $(n-1)^3+n^3+(n+1)^3=m^3$ Prove that 4 divides $n$

Let a line $\ell$ intersect the line $AB$ at $F$, the sides $AC$ and $BC$ of a triangle $ABC$ at $D$ and $E$, respectively and the internal bisector of the angle $BAC$ at $P$. Suppose that $F$ is at the opposite side of $A$ with respect to the line $BC$, $CD = CE$ and $P$ is in the interior the triangle $ABC$. Prove that \[FB \cdot FA+CP^2 = CF^2 \iff AD \cdot BE = PD^2.\]

The bisector of the interior angle at the vertex $B$ of the triangle $ABC$ and the perpendicular line on side $BC$ passing through the vertex $C$ intersects at $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $BD$, respectively, with $N$ on the side $AC$. Find all possibilities of the angles of the triangles $ABC$, if it is known that $\frac{| AM |}{| BC |}=\frac{|CD|}{|BD|}$. .

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is $$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$ Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.

Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$ for all positive $ x$, $ y$, $ z$.

Solve in prime numbers: $ \{\begin{array}{c}\ \ 2a - b + 7c = 1826 \ 3a + 5b + 7c = 2007 \end{array}$

There are two different versions of this problem, each with different solutions. You must find bounds for each of these problems. [list] [*] Alice and Bob are playing a collaborative game in which they agree upon an encoding/strategy. Alice shuffles a deck of 52 cards (numbered 1-52) (*) and takes the first $n$ cards off the top of the deck. Alice then chooses one card to be put to the side, and chooses an ordering of the other $n-1$ cards. Bob then walks into the room seeing the $n-1$ cards in the order Alice put them in. For example, if Alice was given 9-4-10-51-7-8 and chose to put 8 to the side, she could put 5 cards in order of 9-4-10-51-7 which Bob would see. Find the lowest $n$ for which Bob could guarantee that he could use Alice's encoding to find the card placed to the side. (*) (If you would prefer, you can write your solution in terms of the deck being a standard deck with 4 suits and 13 ranks, there's a way to move between them that we can handle.) [*] Alice and Bob are playing a collaborative game in which they agree upon an encoding/strategy. Alice shuffles a deck of 52 cards (numbered 1-52) and takes the first $n$ cards off the top of the deck. Alice then chooses one card to be put to the side, and chooses an ordering of the other $n-1$ cards. Carol then comes in to the room and randomly discards one of the $n-1$ cards, and places the cards back in a way that preserves the order Alice had. Bob then walks into the room seeing the $n-2$ cards in the order Alice put them in. For example, if Alice was given 1-2-3-4-5-6 and chose to put 6 to the side, she would put 5 cards in order of 1-2-3-4-5, and Carol removed 4, Bob would see 1-2-3-5. Find the lowest $n$ for which Bob could guarantee that he could use Alice's encoding to find the card placed to the side. [/list] [i]Proposed by Eric Oh[/i]

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

For a positive integer $n$ denote $A = \{1,2,..., n\}$. Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$. How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$?

A class consisting of $24$ students has participated in the first round of the Georg Mohr Contest, where one could obtain between $0$ and $20$ points. Three of the students obtained exactly the class’s average. If each of the students that scored below the average had scored $4$ points more, the average would have been $3$ points higher. How many students scored above the class’s average?

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

What is the smallest possible volume to surface ratio of a solid cone with height = $1$ unit?