Found problems: 85335
Let f$(x)$ be a continuous function over the real numbers such that for every integer $n$, $f(n) = n^2$ and $f(x) $ is linear over the interval $[n, n + 1]$. There exists a unique two-variable polynomial $g$ such that $g(x, \lfloor x \rfloor) = f(x)$ for all $x$. Compute $g(20, 23)$. (Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. For example, $\lfloor 2\rfloor = 2$ and $\lfloor -3.5 \rfloor = -4$.)
The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$, while the diagonals $AC$ and $BD$ in the point $F$. The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$. Prove that $\angle EHF=90^\circ$.
For a certain real number $x$, the differences between $x^{1919}$, $x^{1960}$ and $x^{2001}$ are all integers. Prove that $x$ is an integer.
Let the real numbers $a$, $b$, and $c$ satisfy the equations
$(a+b)(b+c)(c+a)=abc$ and $(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9$.
Prove that at least one of $a$, $b$, and $c$ equals $0$.
Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$?
Remark: Two cells are called connected if they have a common edge.
There are finitely many primes dividing the numbers $\{ a \cdot b^n + c\cdot d^n : n=1, 2, 3,... \}$ where $a, b, c, d$ are positive integers. Prove that $b=d$.
Suppose that
\[S_n=\frac 59 \times \frac{14}{20} \times \frac{27}{35} \times \cdots \times \frac{2n^2-n-1}{2n^2+n-1}\]
Find $\lim_{n \to \infty} S_n.$
Let $ABC$ be an acute triangle with $AC\neq BC$, $M$ the midpoint of side $AB$, $H$ is the orthocenter of $\triangle ABC$, $D$ on $BC$ is the foot of the altitude from $A$ and $E$ on $AC$ is the foot of the perpendicular from $B$. Prove that the lines $AB, DE$ and the perpendicular to $MH$ through $C$ are concurrent.
Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers .
Prove that if $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$ , then $m \cdot n = p \cdot q$.
Four coplanar regular polygons share a common vertex but have no interior points in common. Each polygon is adjacent to two of the other polygons, and each pair of adjacent polygons has a common side length of $1$. How many possible perimeters are there for all such configurations?
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) more than }5$
Find the sum of the squares of all solutions to $$(x^2 + 7x + 9)^2 = 9.$$
[i]Lightning 1.1[/i]
An acute triangle $\triangle{ABC}$ is inscribed in a circle with centre $O$. The altitudes of the triangle are $AD,BE$ and $CF$. The line $EF$ cut the circumference on $P$ and $Q$.
a) Show that $OA$ is perpendicular to $PQ$.
b) If $M$ is the midpoint of $BC$, show that $AP^2=2AD\cdot{OM}$.
$ABCDE$ is convex pentagon. $m(\widehat{B})=m(\widehat{D})=90^\circ$, $m(\widehat{C})=120^\circ$, $|AB|=2$, $|BC|=|CD|=\sqrt3$, and $|ED|=1$. $|AE|=?$
$ \textbf{(A)}\ \frac{3\sqrt3}{2}
\qquad\textbf{(B)}\ \frac{2\sqrt3}{3}
\qquad\textbf{(C)}\ \frac{3}{2}
\qquad\textbf{(D)}\ \sqrt3 - 1
\qquad\textbf{(E)}\ \sqrt3
$
If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$.
Show that the sequence
$\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$
contains infinitely many odd numbers.
If $\sum_{i=1}^{\infty} u_{i}^{2}$ and $\sum_{i=1}^{\infty} v_{i}^{2}$ are convergent series of real numbers, prove that
$$\sum_{i=1}^{\infty}(u_{i}-v_{i})^{p}$$
is convergent, where $p\geq 2$ is an integer.
There are $n$ students. Denoted the number of the selections to select two students (with their weights are $a$ and $b$) such that $\left| {a - b} \right| \le 1$ (kg) and $\left| {a - b} \right| \le 2$ (kg) by $A_1$ and $A_2$, respectively. Prove that $A_2<3A_1+n$.
In a school, there are $1000$ students in each year level, from Year $1$ to Year $12$. The school has $12 000$ lockers, numbered from $1$ to $12 000$. The school principal requests that each student is assigned their own locker, so that the following condition is satisfied: For every pair of students in the same year level, the difference between their locker numbers must be divisible by their year-level number. Can the principal’s request be satisfied?
Let $x, y, z$ be nonnegative real numbers with $xy + yz + zx = 1$. Show that:
$$\frac{4}{x + y + z} \le (x + y)(\sqrt3 z + 1).$$
Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions.
Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$. Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$
(A) $\pi/2$
(B) $\pi/4$
(C) $\pi $
(D) $\pi/3$
On a Cartesian plane 101 planes are drawn and all points of intersection are labeled. Is it possible, that for every line, 50 of the points have positive coordinates and 50 of the points have negative coordinates
[I]Proposed by S. Ivanov[/i]
In a certain forest there are at least three crossroads, and for any three crossroads of roads A, B and C there is a road from A to B without passing through C. A deer and a hunter are standing at crossroads of different paths. Is it possible that they can exchange positions without their paths crossing at other points, that are not their initial positions?
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that
\[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}.
\]
The ten-letter code $\text{BEST OF LUCK}$ represents the ten digits $0-9$, in order. What 4-digit number is represented by the code word $\text{CLUE}$?
$\textbf{(A)}\ 8671 \qquad
\textbf{(B)}\ 8672 \qquad
\textbf{(C)}\ 9781 \qquad
\textbf{(D)}\ 9782 \qquad
\textbf{(E)}\ 9872$
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Proposed by United Kingdom