[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$? $ \textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25} $

(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$. (iii) Give a rational number between $11/24$ and $6/13$. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]

Let be a parametric set: $$ \mathcal{F}_{\lambda } =\left\{ f:[1,\infty)\longrightarrow\mathbb{R}\bigg| x\in(1,\infty )\implies \int_{x}^{x^2+\lambda^2 x} f\left( \xi\right) d\xi =1\right\} . $$ [b]a)[/b] Show that $ \mathcal{F}_0 =\emptyset . $ [b]b)[/b] Prove that $ \lambda\neq 0 $ implies $ \mathcal{F}_{\lambda }\neq\emptyset . $

Let $ \mathcal{F}$ be a surface which is simply covered by two systems of geodesics such that any two lines belonging to different systems form angles of the same opening. Prove that $ \mathcal{F}$ can be developed (that is, isometrically mapped) into the plane.

Let $a,\ b$ be given positive constants. Evaluate \[\int_0^1 \frac{\ln\ (x+a)^{x+a}(x+b)^{x+b}}{(x+a)(x+b)}dx.\] Own

Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.

In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate the following definite integral. \[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]

Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $4\dfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant rate of $3\dfrac{1}{4}$ miles per hour for the first hour, at $3\dfrac{3}{4}$ miles per hour for the second hour, and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is: $\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 2$

Let $n \geq 0$ be an integer and $f: [0, 1] \rightarrow \mathbb{R}$ an integrable function such that: $$\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1$$ Prove that: $$\int_0^1f(x)^2dx \geq (n+1)^2$$

Evaluate the following definite integral. \[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]

Your team receives up to $100$ points total for the team round. To play this minigame for up to $10$ bonus points, you must decide how to construct an optimal army with number of soldiers equal to the points you receive. Construct an army of $100$ soldiers with $5$ flanks; thus your army is the union of battalions $B_1$, $B_2$, $B_3$, $B_4$, and $B_5$. You choose the size of each battalion such that $|B_1|+|B_2|+|B_3|+|B_4|+|B_5|=100$. The size of each batallion must be integral and non-negative. Then, suppose you receive $n$ points for the Team Round. We will then "supply" your army as follows: if $n>B_1$, we fill in battalion $1$ so that it has $|B_1|$ soldiers; then repeat for the next battalion with $n-|B_1|$ soldiers. If at some point there are not enough soldiers to fill the battalion, the remainder will be put in that battalion and subsequent battalions will be empty. (Ex: suppose you tell us to form battalions of size $\{20,30,20,20,10\}$, and your team scores $73$ points. Then your battalions will actually be $\{20,30,20,3,0\}$.) Your team's army will then "fight" another's. The $B_i$ of both teams will be compared with the other $B_i$, and the winner of the overall war is the army who wins the majority of the battalion fights. The winner receives $1$ victory point, and in case of ties, both teams receive $\tfrac12$ victory points. Every team's army will fight everyone else's and the team war score will be the sum of the victory points won from wars. The teams with ranking $x$ where $7k\leq x\leq 7(k+1)$ will earn $10-k$ bonus points. For example: Team Princeton decides to allocate its army into battalions with size $|B_1|$, $|B_2|$, $|B_3|$, $|B_4|$, $|B_5|$ $=$ $20$, $20$, $20$, $20$, $20$. Team MIT allocates its army into battalions with size $|B_1|$, $|B_2|$, $|B_3|$, $|B_4|$, $|B_5|$ $=$ $10$, $10$, $10$, $10$, $60$. Now suppose Princeton scores $80$ points on the Team Round, and MIT scores $90$ points. Then after supplying, the armies will actually look like $\{20, 20, 20, 20, 0\}$ for Princeton and $\{10, 10, 10, 10, 50\}$ for MIT. Then note that in a war, Princeton beats MIT in the first four battalion battles while MIT only wins the last battalion battle; therefore Princeton wins the war, and Princeton would win $1$ victory point.

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

Calculate the following indefinite integrals. (1) $ \int \frac {3x \plus{} 4}{x^2 \plus{} 3x \plus{} 2}dx$ (2) $ \int \sin 2x\cos 2x\cos 4x\ dx$ (3) $ \int xe^{x}dx$ (4) $ \int 5^{x}dx$

In a plane, there is an infinite triangular grid consists of equilateral triangles whose lengths of the sides are equal to $ 1$, call the vertices of the triangles the lattice points, call two lattice points are adjacent if the distance between the two points is equal to $ 1;$ A jump game is played by two frogs $ A,B,$ "A jump" is called if the frogs jump from the point which it is lying on to its adjacent point, " A round jump of $ A,B$" is called if first $ A$ jumps and then $ B$ by the following rules: Rule (1): $ A$ jumps once arbitrarily, then $ B$ jumps once in the same direction, or twice in the opposite direction; Rule (2): when $ A,B$ sits on adjacent lattice points, they carry out Rule (1) finishing a round jump, or $ A$ jumps twice continually, keep adjacent with $ B$ every time, and $ B$ rests on previous position; If the original positions of $ A,B$ are adjacent lattice points, determine whether for $ A$ and $ B$,such that the one can exactly land on the original position of the other after a finite round jumps.

Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.

Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

For a fixed natural number $m \geq 2$, prove that [b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\] [b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.

Let $S_n$ be the area of the figure enclosed by a curve $y=x^2(1-x)^n\ (0\leq x\leq 1)$　and the $x$-axis. Find $\lim_{n\to\infty} \sum_{k=1}^n S_k.$

Find all functions $ f:[0,1]\longrightarrow [0,1] $ that are bijective, continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow\mathbb{R} , $ the following equality holds. $$ \int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx $$

A sequence $a_n$ satisfies $a_1 =2t-3$ ($t \ne 1,-1$), and $a_{n+1}=\dfrac{(2t^{n+1}-3)a_n+2(t-1)t^n-1}{a_n+2t^n-1}$. [list] [b][i]i)[/i][/b] Find $a_n$, [b][i]ii)[/i][/b] If $t>0$, compare $a_{n+1}$ with $a_n$.[/list]

\[\int_0^{\frac \pi 4} \cot(x)\sqrt{\sin(x)}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number. 1972 Tokyo University of Education entrance exam

Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.