Found problems: 85335
Juanita wrote the numbers from $1$ to $13$ , calculated the sum of all the digits he had written and obtained
$$1+2+3+4+5+6+7+8+9+(1+0)+(1+1)+(1+2)+(1+3)=55.$$
His brother Ariel wrote the numbers from $1$ to $100$ and calculated the sum of all the digits written. Find the value of Ariel's sum.
In triangle $\vartriangle ABC$, we have marked points $A_1$ on side $BC, B_1$ on side $AC$, and $C_1$ on side $AB$ so that $AA_1$ is an altitude, $BB_1$ is a median, and $CC_1$ is an angle bisector. It is known that $\vartriangle A_1B_1C_1$ is equilateral. Prove that $\vartriangle ABC$ is equilateral too.
(Note: A median connects a vertex of a triangle with the midpoint of the opposite side. Thus, for median $BB_1$ we know that $B_1$ is the midpoint of side $AC$ in $\vartriangle ABC$.)
Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.
Let $x_i\ (i = 1, 2, \cdots 22)$ be reals such that $x_i \in [2^{i-1},2^i]$. Find the maximum possible value of
$$(x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}})$$
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
Let $\alpha$ and $\beta$ be irrational numbers with the property that $$\frac{1}{\alpha} +\frac{1}{\beta}=1$$
Let$\{a_n\}$ and $\{b_n\}$ be the sequences given by $a_n= \lfloor n\alpha \rfloor$ and $b_n= \lfloor n\beta \rfloor$ respectively. Prove that the sequences $\{ a_n\}$ and $\{ b_n \} $ has no term in common and cover all the natural numbers.
I know this theorem from long ago, but forgot the proof of it. Can anybody help me with this?
Positive integer $n$ when divided with number $3$ gives remainder $a$, when divided with $5$ has remainder $b$ and when divided with $7$ gives remainder $c$. Find remainder when dividing number $n$ with $105$ if $4a+3b+2c=30$
Prove that there is no positive rational number $x$ such that \[x^{\lfloor x\rfloor }=\frac{9}{2}.\]
In the $xy$-plane, how many lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer pass through the point $(4,3)$?
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$
Numbers from$ 1$ to $8$ were written at the vertices of the cube, and on each edge the absolute value of the difference between the numbers at its ends.. What is the smallest number of different numbers that can be written on the edges?
Is it possible to cut the table of size $2007\times2007$ into figures shown here, if one has to use
at least one figure of each sort? $\begin{picture}(45,25) \put(5,5){\put(0,0){\line(1,0){16}}\put(0,8){\line(1,0){24}}\put(0,16){\line(1,0){24}}\put(8,24){\line(1,0){16}}\put(0,0){\line(0,1){16}}\put(8,0){\line(0,1){24}}\put(16,0){\line(0,1){24}}\put(24,8){\line(0,1){16}}}\put(35,5){\put(0,0){\line(1,0){8}}\put(0,8){\line(1,0){8}}\put(0,16){\line(1,0){8}}\put(0,24){\line(1,0){8}}\put(0,0){\line(0,1){24}}\put(8,0){\line(0,1){24}}}\end{picture}$
How many integer triples $(x,y,z)$ are there such that \[\begin{array}{rcl} x - yz^2&\equiv & 1 \pmod {13} \\ xz+y&\equiv& 4 \pmod {13} \end{array}\] where $0\leq x < 13$, $0\leq y <13$, and $0\leq z< 13$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 23
\qquad\textbf{(C)}\ 36
\qquad\textbf{(D)}\ 49
\qquad\textbf{(E)}\ \text{None of above}
$
Positive real numbers $a$,$b$,$c$ are given such that $abc=1$.Prove that
$$2(a+b+c)+\frac{9}{(ab+bc+ca)^2}\geq7.$$
John had to solve a math problem in the class. While cleaning the blackboard, he accidentally erased a part of his problem as well: the text that remained on board was $37 \cdot(72 + 3x) = 14**45$, where $*$ marks an erased digit. Show that John can still solve his problem, knowing that $x$ is an integer
A function $f:[0,\infty)\to[0,\infty)$ is integrable and
$$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$
Prove the following inequality.
$$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$
Zandrew Hao has $n^2$ dollars, where $n$ is an integer. He is a massive fan of the singer Pachary Zerry, and he wants to buy many copies of his $3$ albums, which cost $\$8$, $\$623$, and $\$835$ (two of them are very rare). Find the sum of the $3$ greatest values of $n$ such that Zandrew can't spend all of his money on albums.
Let $ABCD$ be a quadrilateral (with non-perpendicular diagonals).
The perpendicular from $A$ to $BC$ meets $CD$ at $K$.
The perpendicular from $A$ to $CD$ meets $BC$ at $L$.
The perpendicular from $C$ to $AB$ meets $AD$ at $M$.
The perpendicular from $C$ to $AD$ meets $AB$ at $N$.
1. Prove that $KL$ is parallel to $MN$.
2. Prove that $KLMN$ is a parallelogram if $ABCD$ is cyclic.
Let $N$ be the orthocentre of triangle $ABC$ (i .e. the point where the altitudes meet). Prove that the circumscribed circles of triangles $ABN, ACN$ and $BCN$ each have equal radius.
Let $a, b, c$ and $d$ be arbitrary real numbers. Prove that if $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ then $a=b=c=d$.
Let $AB$ be a segment of a plane. Is it possible to paint the plane in $2009$ colors in such a way that both of the following conditions are satisfied?
1) Every two points of the same color can be connected by a polygonal line.
2) For any point $C$ of $AB$, every $n \in N$ and every $k\in \{1,2,3,...,2009\}$ , there exists a point $D$, painted in $k$-th color such that the length of $CD$ is less than $0,0...01$, where all the zeros after the decimal point are exactly $n$.
In a triangle $A_1A_2A_3$, the excribed circles corresponding to sides $A_2A_3$, $A_3A_1$, $A_1A_2$ touch these sides at $T_1$, $T_2$, $T_3$, respectively. If $H_1$, $H_2$, $H_3$ are the orthocenters of triangles $A_1T_2T_3$, $A_2T_3T_1$, $A_3T_1T_2$, respectively, prove that lines $H_1T_1$, $H_2T_2$, $H_3T_3$ are concurrent.
Gary plays the following game with a fair $n$-sided die whose faces are labeled with the positive integers between $1$ and $n$, inclusive: if $n = 1$, he stops; otherwise he rolls the die, and starts over with a $k$-sided die, where $k$ is the number his $n$-sided die lands on. (In particular, if he gets $k = 1$, he will stop rolling the die.) If he starts out with a $6$-sided die, what is the expected number of rolls he makes?
[b]p1.[/b] a) There are barrels weighing $1, 2, 3, 4, ..., 19, 20$ pounds. Is it possible to distribute them equally (by weight) into three trucks?
b) The same question for barrels weighing $1, 2, 3, 4, ..., 9, 10$ pounds.
[b]p2.[/b] There are apples and pears in the basket. If you add the same number of apples there as there are now pears (in pieces), then the percentage of apples will be twice as large as what you get if you add as many pears to the basket as there are now apples. What percentage of apples are in the basket now?
[b]p3.[/b] What is the smallest number of integers from $1000$ to $1500$ that must be marked so that any number $x$ from $1000$ to $1500$ differs from one of the marked numbers by no more than $10\% $of the value of $x$?
[b]p4.[/b] Draw a perpendicular from a given point to a given straight line, having a compass and a short ruler (the length of the ruler is significantly less than the distance from the point to the straight line; the compass reaches from the point to the straight line “with a margin”).
[b]p5.[/b] There is a triangle on the chessboard (left figure). It is allowed to roll it around the sides (in this case, the triangle is symmetrically reflected relative to the side around which it is rolled). Can he, after a few steps, take the position shown in right figure?
[img]https://cdn.artofproblemsolving.com/attachments/f/5/eeb96c92f30b837e7ed2cdf7cf77b0fbb8ceda.png[/img]
[b]p6.[/b] The natural number $a$ is less than the natural number $b$. In this case, the sum of the digits of number $a$ is $100$ less than the sum of the digits of number $b$. Prove that between the numbers $ a$ and $b$ there is a number whose sum of digits is $43$ more than the sum of the digits of $a$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.
Evaluate the integrals as follows.
(1) $\int \frac{x^2}{2-x}\ dx$
(2) $\int \sqrt[3]{x^5+x^3}\ dx$
(3) $\int_0^1 (1-x)\cos \pi x\ dx$