Found problems: 85335
For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result?
(G Galperin)
Find the necessary and sufficient conditions for two conics that every tangent to one of them contains a real point of the other.
Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$ and the graphs of the functions $y =\sqrt{x^3+ 1}$ and $y = - \sqrt[3]{x^2+ 2x}$.
Let $APBCQ$ be a cyclic pentagon. A point $M$ inside triangle $ABC$ is such that $\angle MAB = \angle MCA$, $\angle MAC = \angle MBA$ and $\angle PMB = \angle QMC = 90^\circ$. Prove that $AM$, $BP$, and $CQ$ concur.
[i]Anant Mudgal and Navilarekallu Tejaswi[/i]
Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].
Function $f(x)=\frac{x}{1-2^x}-\frac{x}{2}$ is
$\text{(A)}$ an even function, not an odd function.
$\text{(B)}$ an odd function, not an even function.
$\text{(C)}$ an even function, also an odd function.
$\text{(D)}$ neither an even function, nor an odd function.
Let $n$ be the number of isosceles triangles whose vertices are also the vertices of a regular 2019-gon.
Then the remainder when $n$ is divided by 100
[list=1]
[*] 15
[*] 25
[*] 35
[*] 65
[/list]
Let $z$ be a positive integer that is not divisible by $8$. Furthermore, let $n \geqslant 2$ be a positive integer. Prove that none of the numbers of the form $z^n + z + 1$ is a square number.
[i](Walther Janous)[/i]
A utility company is building a network to send electricity to fifty houses, with addresses $0, 1, 2, \ldots , 49. $ The power center only connects directly to house $0$, so electricity reaches all other houses through a system of wires that connects specific pairs of houses. To save money, the company only lays wires between as few pairs of distinct houses as possible; additionally, two houses with addresses $a$ and $b$ can only have a wire between them if at least one of the following three conditions is met:
[list]
[*]$10$ divides both $a$ and $b.$
[*]$\lfloor \tfrac{a}{10} \rfloor \equiv \lfloor \tfrac{b}{10}\rfloor \pmod{5}.$
[*]$\lceil \tfrac{a}{10} \rceil\equiv \lceil \tfrac{b}{10}\rceil\pmod{5}.$
[/list]
Letting $N$ be the number of distinct ways such a wire system can be configured so that every house receives electricity , find the remainder when $N$ is divided by $1000.$
Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$.
The eyes of a magician are blindfolded while a person $A$ from the audience arranges $n$ identical coins in a row, some are heads and the others are tails. The assistant of the magician asks $A$ to write an integer between $1$ and $n$ inclusive and to show it to the audience. Having seen the number, the assistant chooses a coin and turns it to the other side (so if it was heads it becomes tails and vice versa) and does not touch anything else. Afterwards, the bandages are removed from the magician, he sees the sequence and guesses the written number by $A$. For which $n$ is this possible?
[hide=Spoiler hint] The original formulation asks: a) Show that if $n$ is possible, so is $2n$; b) Show that only powers of $2$ are possible; I have omitted this from the above formulation, for the reader's interest. [/hide]
A unit square is divided into two rectangles in such a way that the smaller rectangle can be put on the greater rectangle with every vertex of the smaller on exactly one of the edges of the greater. Calculate the dimensions of the smaller rectangle.
Let $a,b,c,d$ be real numbers such that
\[a+b+c+d=-2\]
\[ab+ac+ad+bc+bd+cd=0\]
Prove that at least one of the numbers $a,b,c,d$ is not greater than $-1$.
Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.
Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.
$A_1A_2A_3A_4$ is a tangential quadrilateral with perimeter $p_1$ and sum of the diagonals $k_1$ .$B_1B_2B_3B_4$ is a tangential quadrilateral with perimeter $p_2$ and sum of the diagonals $k_2$ .Prove that $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are congruent squares if $$ p_1^2+p_2^2=(k_1+k_2)^2 $$
On each side of triangle $ABC$, two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors.
a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors.
b) Solve p.a) drawing only three lines.
The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal.
Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.
The game involves two players $A$ and $B$. Player $A$ sets the value of one of the coefficients $a, b$ or $c$ of the polynomial $$x^3 + ax^2 + bx + c.$$
Player $B$ indicates the value of any of the two remaining coefficients . Player $A$ then sets the value of the last coefficients. Is there a strategy for player A such that no matter how player $B$ plays, the equation $$x^3 + ax^2 + bx + c = 0$$ to have three different (real) solutions?
Define the sequence $x_1,x_2,\dots$ by the initial terms $x_1=2, x_2=4$, and the recurrence relation
\[x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n} \quad \text{for} \quad n \ge 1.\]
Prove that $\lim_{n \to \infty} \frac{x_n}{2^n}$ exists and satisfies
\[\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.\]
Suppose that $ a_1,\cdots , a_{25}$ are non-negative integers, and $ k$ is the smallest of them. Prove that
$$\big[\sqrt{a_1}\big]+\big[\sqrt{a_2}\big]+\cdots+\big[\sqrt{a_{25}}\big ]\geq\big[\sqrt{a_1+a_2+\cdots+a_{25}+200k}\big].$$
(As usual, $[x]$ denotes the integer part of the number $x$ , that is, the largest integer not exceeding $x$.)
The set $M=\{1,2,\ldots,2007\}$ has the following property: If $n$ is an element of $M$, then all terms in the arithmetic progression with its first term $n$ and common difference $n+1$, are in $M$. Does there exist an integer $m$ such that all integers greater than $m$ are elements of $M$?
At $n$ distinct points of a circular race course there are $n$ cars ready to start. Each car moves at a constant speed and covers the circle in an hour. On hearing the initial signal, each of them selects a direction and starts moving immediately. If two cars meet, both of them change directions and go on without loss of speed. Show that at a certain moment each car will be at its starting point.
An object starting from rest can roll without slipping down an incline.
Which of the following four objects, each a uniform solid sphere released from rest, would have the largest speed after the center of mass has moved through a vertical distance h?
$\textbf{(A)}\text{a sphere of mass M and radius R}$
$\textbf{(B)}\text{a sphere of mass 2M and radius} \frac{R}{2}$
$\textbf{(C)}\text{a sphere of mass }\frac{M}{2} \text{ and radius 2R}$
$\textbf{(D)}\text{a sphere of mass 3M and radius 3R}$
$\textbf{(E)}\text{All objects would have the same speed}$
An infinite, strictly increasing sequence $\{a_n\}$ of positive integers satisfies the condition $a_{a_n}\le a_n + a_{n + 3}$ for all $n\ge 1$. Prove that there are infinitely many triples $(k, l, m)$ of positive integers such that $k <l <m$ and $a_k + a_m = 2a_l$.