Found problems: 85335
Let a$,b,c,d$ and $x,y,z,t$ be real numbers such that
$0\le a,b,c,d \le 1$ , $x,y,z,t \ge 1$ and $a+b+c+d +x+y+z+t=8$.
Prove that $a^2+b^2+c^2+d^2+x^2+y^2+z^2+t^2\le 28$
Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$.
(a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$.
(b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.
Are there infinitely many positive integers $n$ that can not be represented as $n = a^3+b^5+c^7+d^9+e^{11}$, where $a,b,c,d,e$ are positive integers? Explain why.
$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?
$ \textbf{(A)}\ 1404 \qquad \textbf{(B)}\ 1462 \qquad \textbf{(C)}\ 1604 \qquad \textbf{(D)}\ 1605 \qquad \textbf{(E)}\ 1804$
Find all values of $a$ such that $x^6 - 6x^5 + 12x^4 + ax^3 + 12x^2 - 6x +1$ is nonnegative for all real $x$.
For which $n\ge 3$ does there exist positive integers $a_1<a_2<\cdots <a_n$, such that: $$a_n=a_1+...+a_{n-1}, \hspace{0.5cm} \frac{1}{a_1}=\frac{1}{a_2}+...+\frac{1}{a_n}$$ are both true?
[i]Proposed by Ivan Chan Kai Chin[/i]
Determine all pairs $(h,s)$ of positive integers with the following property:
If one draws $h$ horizontal lines and another $s$ lines which satisfy
(i) they are not horizontal,
(ii) no two of them are parallel,
(iii) no three of the $h + s$ lines are concurrent,
then the number of regions formed by these $h + s$ lines is 1992.
What is the units digit of $ 13^{2003}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9$
In a cyclic quadrilateral $ ABCD$ the sides $ AB$ and $ AD$ are equal, $ CD>AB\plus{}BC$. Prove that $ \angle ABC>120^\circ$.
Let $n>1$ be a positive integer. Call a rearrangement $a_1,a_2, \cdots , a_n$ of $1,2, \cdots , n$ [i]nice[/i] if for every $k = 2,3, \cdots , n$, we have that $a_1 + a_2 + \cdots + a_k$ is not divisible by $k$.
(a) If $n>1$ is odd, prove that there is no nice arrangement of $1,2, \cdots , n$.
(b) If $n$ is even, find a [i]nice[/i] arrangement of $1,2, \cdots , n$.
Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.
In a plane, equilateral triangle $ABC$, square $BCDE$, and regular dodecagon $DEFGHIJKLMNO$ each have side length 1 and do not overlap. Find the area of the circumcircle of $\triangle AFN$.
Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$.
Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$.
Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$.
(Hungary)
Find the number of 8-digit base-6 positive integers $(a_1a_2a_3a_4a_5a_6a_7a_8)_6$ (with leading zeros permitted) such that $(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6$ for $i=1,2,\ldots,7$, where indices are taken modulo $8$ (so $a_9=a_1$, $a_{10}=a_2$, and so on).
[i]Victor Wang[/i]
An equilateral triangle $ABC$ is given. On the line through $B$ parallel to $AC$ there is a point $D$, such that $D$ and $C$ are on the same side of the line $AB$. The perpendicular bisector of $CD$ intersects the line $AB$ in $E$. Prove that triangle $CDE$ is equilateral.
Let $ABC$ be a triangle, $I$ and $O$ be its incenter and circumcenter respectively. $A'$ is symmetric to $O$ with respect to line $AI$. Points $B'$ and $C'$ are defined similarly. Prove that the nine-point centers of triangles $ABC$ and $A'B'C'$ coincide.
Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$
A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point.
[b]Dan Schwarz[/b]
It is given a plane with a coordinate system with a beginning at the point $O$. $A(n)$, when $n$ is a natural number is a count of the points with whole coordinates which distances to $O$ are less than or equal to $n$.
(a) Find
$$\ell=\lim_{n\to\infty}\frac{A(n)}{n^2}.$$
(b) For which $\beta$ $(1<\beta<2)$ does the following limit exist?
$$\lim_{n\to\infty}\frac{A(n)-\pi n^2}{n^\beta}$$
$234$ viewers came to the cinema. Determine for which$ n \ge 4$ the viewers could be can be arranged in $n$ rows so that every viewer in $i$-th row gets to know just $j$ viewers in $j$-th row for any $i, j \in \{1, 2,... , n\}, i\ne j$. (The relationship of acquaintance is mutual.)
(Tomáš Jurík)
Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(f(x)+x+f(y)f(z))=f(x+zf(x)+zf(y))-xf(z-1)\]for all $x,y,z\in\mathbb{R}$.
Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.
On a isosceles triangle $\triangle ABC$ with $AB=BC$ let $K,M$ be the midpoints of $AB,AC$ respectivily. Let $(CKB)$ intersect $BM$ at $N \ne M$, the line through $N$ parallel to $AC$ intersects $(ABC)$ at $A_1,C_1$. Show that $\triangle A_1BC_1$ is equilateral.