Let $A=(a_{k,l})_{k,l=1,...,n}$ be a complex $n \times n$ matrix such that for each $m \in \{1,2,...,n\}$ and $1 \leq j_{1} <...<j_{m}$ the determinant of the matrix $(a_{j_{k},j_{l}})_{k,l=1,...,n}$ is zero. Prove that $A^{n}=0$ and that there exists a permutation $\sigma \in S_{n}$ such that the matrix $(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n}$ has all of its nonzero elements above the diagonal.

Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.

Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum $$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots$$ of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A)} \text{ 16} \qquad \textbf{(B)} \text{ 17} \qquad \textbf{(C)} \text{ 19} \qquad \textbf{(D)} \text{ 23} \qquad \textbf{(E)} \text{ 36}$

A set of $n$ positive integers is said to be [i]balanced[/i] if for each integer $k$ with $1 \leq k \leq n$, the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$.

Let $n$ be a positive integer and let $a_1,a_2,...,a_n$ be non-zero positive integers.Prove that $$\sum_{k=1}^n\frac{\sqrt{a_k}}{1+a_1+a_2+...+a_k}<\sum_{k=1}^{n^2}\frac{1}{k}.$$

Let $n\ge 3$ be a positive integer. Amy wrote all the integers from $1$ to $n^2$ on the $n\times n$ grid, so that each cell contains exactly one number. For $i=1,2,\cdots ,n^2-1$, the cell containing $i$ shares a common side with the cell containing $i+1$. Each turn, Bred can choose one cell, and check what number is written. Bred wants to know where $1$ is written by less than $3n$ turns. Determine whether $n$ such that Bred can always achieve his goal is infinite.

Calculate$$\lim \limits_{n \to \infty}\frac{n!\left(1+\frac{1}{n}\right)^{n^2+n}}{n^{n+\frac{1}{2}}}$$

[i]Wind chill[/i] is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation: $$(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),$$ where temperature is measured in degrees Fahrenheit $(^{\circ}\text{F})$ and and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ}\text{F} $ and the wind speed is $18$ mph. Which of the following is closest to the approximate wind chill? $\textbf{(A)}~18\qquad\textbf{(B)}~23\qquad\textbf{(C)}~28\qquad\textbf{(D)}~32\qquad\textbf{(E)}~35$

Radius of the circle $\omega_A$ with centre at vertex $A$ of a triangle $\triangle{ABC}$ is equal to the radius of the excircle tangent to $BC$. The circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other. [i](L. Emelyanov)[/i]

Let $ABC$ be a triangle with $AB = 20$ and $AC = 22$. Suppose its incircle touches $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at $D$, $E$, and $F$ respectively, and $P$ is the foot of the perpendicular from $D$ to $\overline{EF}$. If $\angle BPC = 90^{\circ}$, then compute $BC^2$. [i]Proposed by Ankan Bhattacharya[/i]

Design a planar finite non-empty set $S$ satisfying the following two conditions: (a) every line meets $S$ in at most four points; and (b) every $2$-colouring of $S$ - that is, each point of $S$ is coloured one of two colours - yields (at least) three monochromatic collinear points.

Find the number of eight-digit positive integers that are multiples of $9$ and have all distinct digits.

Let $N_{11}$ be the answer to problem 11. Concave heptagon $HOPKINS$, where $180^\circ<\angle HOP<270^\circ$, has area $N_{11}$, and $HP=NI\sqrt{24}$. Suppose that $HONS$ and $OPKI$ are congruent squares. Compute the common area of each of these squares.

Two squares are arranged as shown. Prove that the area of the black triangle equal to the sum of the gray areas. [img]https://2.bp.blogspot.com/-byhWqNr1ras/XTq-NWusg2I/AAAAAAAAKZA/1sxEZ751v_Evx1ij7K_CGiuZYqCjhm-mQCK4BGAYYCw/s400/Oral%2BSharygin%2B2017%2B8.9%2Bp5.png[/img]

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle $ABC$, the median $m_a$ meets $BC$ at $A'$ and the circumcircle again at $A_1$. The symmedian $s_a$ meets $BC$ at $M$ and the circumcircle again at $A_2$. Given that the line $A_1A_2$ contains the circumcenter $O$ of the triangle, prove that: [i](a) [/i]$\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;$ [i](b) [/i]$1+4b^2c^2 = a^2(b^2+c^2)$

Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

Let $ABC$ be an equilateral triangle. $N$ is a point on the side $AC$ such that $\vec{AC} = 7\vec{AN}$, $M$ is a point on the side $AB$ such that $MN$ is parallel to $BC$ and $P$ is a point on the side $BC$ such that $MP$ is parallel to $AC$. Find the ratio of areas $\frac{ (MNP)}{(ABC)}$

Let $\mathcal{C}$ be the circumference inscribed in the triangle $ABC,$ which is tangent to sides $BC, AC, AB$ at the points $A' , B' , C' ,$ respectively. The distinct points $K$ and $L$ are taken on $\mathcal{C}$ such that $$\angle AKB'+\angle BKA' =\angle ALB'+\angle BLA'=180^{\circ}.$$ Prove that the points $A', B', C'$ are equidistant from the line $KL.$

Let $S = x + y +z$ where $x, y, z$ are three nonzero real numbers satisfying the following system of inequalities: $$xyz > 1$$ $$x + y + z >\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ Prove that $S$ can take on any real values when $x, y, z$ vary

Alice and Bob are playing a game, starting with a binary string$ b$ of length $2022$. In each step, the rightmost digit of the string is deleted. If the deleted digit was $1$, Alice gets to choose which digit she wants to append on the left. Otherwise, Bob gets to choose the digit to append on the left of the string. Alice would like to turn the string $b$ into the all-zero string $\underbrace{00 . . . 0}_{2022}$, in the least number of steps possible, while Bob would like to maximize the number of steps necessary, or prevent Alice from doing this at all. a) Is there a string $b$ for which Bob can prevent Alice in her goal, if both players play optimally? b) If the answer to part a is yes, find all such strings $b$. If the answer is no, find the maximal game time and find the set of strings $b$ for which the game time is maximal.

How many distinct ordered triples $(x,y,z)$ satisfy the equations \begin{align*}x+2y+4z&=12 \\ xy+4yz+2xz&=22 \\ xyz&=6~~?\end{align*} $\textbf{(A) }\text{none}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad \textbf{(E) }6$

$ABC$ is an equilateral triangle, and $ADEF$ is a square. If $D$ lies on side $AB$ and $E$ lies on side $BC$, what is the ratio of the area of the equilateral triangle to the area of the square?

When Walter drove up to the gasoline pump, he noticed that his gasoline tank was $\frac{1}{8}$ full. He purchased $7.5$ gallons of gasoline for $ \$10$. With this additional gasoline, his gasoline tank was then $\frac{5}{8}$ full. The number of gallons of gasoline his tank holds when it is full is $\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5$