How many lattice points $(v, w, x, y, z)$ does a $5$-sphere centered on the origin, with radius $3$, contain on its surface or in its interior?

Consider a rectangular grid of points consisting of $4$ rows and $84$ columns. Each point is coloured with one of the colours red, blue or green. Show that no matter whatever way the colouring is done, there always exist four points of the same colour that form the vertices of a rectangle. An illustration is shown in the figure below.

Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$

Consider all fractions $\tfrac{a}{b}$ where $1\leq b\leq100$ and $0\leq a\leq b$. Of these fractions, let $\tfrac{m}{n}$ be the smallest fraction such that $\tfrac{m}{n}>\tfrac{2}{7}$. What is $\tfrac{m}{n}$?

Let $a,b,c$ be real numbers with $a,c\ne0$. Prove that if $r$ is a real root of $ax^2+bx+c=0$ and $s$ a real root of $-ax^2+bx+c=0$, then there is a root of a $\frac a2x^2+bx+c=0$ between $r$ and $s$.

Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.

Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.

Is it possible that the perimeter of a triangle whose side lengths are integers, is divisible by the double of the longest side length?

a) Find at least two functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ b) Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ Show that $ f(x^3)\geq x^2,$ for all $x \in \mathbb{R}^+.$ Can we find the best constant $a\in \Bbb{R}$ such that $f(x)\geq x^a,$ for all $x \in \mathbb{R}^+?$

In trapezoid $ABCD$ with $AD\parallel BC$, $AB=6$, $AD=9$, and $BD=12$. If $\angle ABD=\angle DCB$, find the perimeter of the trapezoid.

Prove or disprove the existence of a function $f : S \to R$ such that for all $x \ne y \in S$ we have $|f(x) - f(y)| \ge \frac{1}{x^2 + y^2}$, in each of the cases: a) $S = R$ b) $S = Q$.

Given $T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1$ for $n>0.$ Prove: (i) If $m \ne n,$ $T_m$ and $T_n$ have no common factor greater than $1.$ (ii) $\sum_{i=1}^{\infty} \frac{1}{T_i }=1.$

Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers? $\textbf{(A)}\hspace{.05in}76531 \qquad \textbf{(B)}\hspace{.05in}86724 \qquad \textbf{(C)}\hspace{.05in}87431 \qquad \textbf{(D)}\hspace{.05in}96240 \qquad \textbf{(E)}\hspace{.05in}97403 $

Let $BM$ be the median of the triangle $ABC$, in which $AB> BC$. Point $P$ is chosen so that $AB \parallel PC$ and$ PM \perp BM$. The point $Q$ is chosen on the line $BP$ so that $\angle AQC = 90^o$, and the points $B$ and $Q$ lie on opposite sides of the line $AC$. Prove that $AB = BQ$. (Mikhail Standenko)

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a}{a+\sqrt{(a+b)(a+c)}}+\frac{b}{b+\sqrt{(b+c)(b+a)}}+\frac{c}{c+\sqrt{(c+a)(c+b)}}\leq\frac{2a^2+ab}{(b+\sqrt{ca}+c)^2}+\frac{2b^2+bc}{(c+\sqrt{ab}+a)^2}+\frac{2c^2+ca}{(a+\sqrt{bc}+b)^2}.$$

Finitely many cells of an infinite square board are colored black. Prove that one can choose finitely many squares in the plane of the board so that the following conditions are satisfied: (i) The interiors of any two different squares are disjoint; (ii) Each black cell lies in one of these squares; (iii) In each of these squares, the black cells cover at least $\frac15$ and at most $\frac45$ of the area of that square.

An $ 8$-foot by $ 10$-foot floor is tiled with square tiles of size $ 1$ foot by $ 1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $ 1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); fill(unitsquare,gray); filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black); filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black); filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black); filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);[/asy]$ \textbf{(A)}\ 80\minus{}20\pi \qquad \textbf{(B)}\ 60\minus{}10\pi \qquad \textbf{(C)}\ 80\minus{}10\pi \qquad \textbf{(D)}\ 60\plus{}10\pi \qquad \textbf{(E)}\ 80\plus{}10\pi$

Let $a$ and $b$ be non-negative integers. Consider a sequence $s_1$, $s_2$, $s_3$, $. . .$ such that $s_1 = a$, $s_2 = b$, and $s_{i+1} = |s_i - s_{i-1}|$ for $i \ge 2$. Prove that there is some $i$ for which $s_i = 0$.

Which is the best value for the coefficient of friction between the block and the surface? (a) $\text{0.05}$ (b) $\text{0.07}$ (c) $\text{0.09}$ (d) $\text{0.5}$ (e) $\text{0.6}$

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

A [i]complex set[/i], along with its [i]complexity[/i], is defined recursively as the following: [list] [*]The set $\mathbb{C}$ of complex numbers is a complex set with complexity $1$. [*]Given two complex sets $C_1, C_2$ with complexity $c_1, c_2$ respectively, the set of all functions $f:C_1\rightarrow C_2$ is a complex set denoted $[C_1, C_2]$ with complexity $c_1 + c_2$. [/list] A [i]complex expression[/i], along with its [i]evaluation[/i] and its [i]complexity[/i], is defined recursively as the following: [list] [*]A single complex set $C$ with complexity $c$ is a complex expression with complexity $c$ that evaluates to itself. [*]Given two complex expressions $E_1, E_2$ with complexity $e_1, e_2$ that evaluate to $C_1$ and $C_2$ respectively, if $C_1 = [C_2, C]$ for some complex set $C$, then $(E_1, E_2)$ is a complex expression with complexity $e_1+e_2$ that evaluates to $C$. [/list] For a positive integer $n$, let $a_n$ be the number of complex expressions with complexity $n$ that evaluate to $\mathbb{C}$. Let $x$ be a positive real number. Suppose that \[a_1+a_2x+a_3x^2+\dots = \dfrac{7}{4}.\] Then $x=\frac{k\sqrt{m}}{n}$, where $k$,$m$, and $n$ are positive integers such that $m$ is not divisible by the square of any integer greater than $1$, and $k$ and $n$ are relatively prime. Compute $100k+10m+n$. [i]Proposed by Luke Robitaille and Yannick Yao[/i]

Determine all functions $f : R^+ \to R^+$, so that for all $x, y > 0$: $$f(xy) \le \frac{xf(y) + yf(x)}{2}$$

Show that $ \frac{\frac{1}{1\cdot 2} +\frac{1}{3\cdot 4}+\cdots +\frac1{1997\cdot 1998}}{\frac{2}{1000\cdot 1998} +\frac{1}{1001\cdot 1997}} $ is an integer number. [i]Bogdan Enescu[/i]

Let $ ABCDA’B’C’D’ $ a right parallelepiped and $ M,N $ the feet of the perpendiculars of $ BD $ through $ A’, $ respectively, $ C’. $ We know that $ AB=\sqrt 2, BC=\sqrt 3, AA’=\sqrt 2. $ [b]a)[/b] Prove that $ A’M\perp C’N. $ [b]b)[/b] Calculate the dihedral angle between the plane formed by $ A’MC $ and the plane formed by $ ANC’. $

Are there four real numbers $a, b, c, d$ for every three positive real numbers $x, y, z$ with the property $ad + bc = x$, $ac + bd = y$, $ab + cd = z$ and one of the numbers $a, b, c, d$ is equal to the sum of the other three?