Found problems: 85335
Given a natural number $ n, $ prove that the set $ \{ -n+1,-n+2,\ldots ,-1,1,2,\ldots ,n-1,n\} $ can be partitioned into $ k $ subsets such that the sums of all elements of each of these subsets are equal, if and only if $ n $ is multiple of $ k. $
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.
How many ways are there to put the cards in the three boxes so that the trick works?
All of the digits of a seven-digit positive integer are either $7$ or $8.$ If this integer is divisible by $9,$ what is the sum of its digits?
Let $\mathcal{C}$ be a right circular cone with height $\sqrt{15}$ and base radius $1$. Let $V$ be the vertex of $\mathcal{C}$, $B$ be a point on the circumference of the base of $\mathcal{C}$, and $A$ be the midpoint of $VB$. An ant travels at constant velocity on the surface of the cone from $A$ to $B$ and makes two complete revolutions around $\mathcal{C}$. Find the distance the ant travelled.
In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. if $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, Where $m,n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.
Let $n$ be a positive integer and let $P(x)$ be a monic polynomial of degree $n$ with real coefficients. Also let $Q(x)=(x+1)^2(x+2)^2\dots (x+n+1)^2$. Consider the minimum possible value $m_n$ of $\displaystyle\sum_{i=1}^{n+1} \dfrac{i^2P(i^2)^2}{Q(i)}$. Then there exist positive constants $a,b,c$ such that, as $n$ approaches infinity, the ratio between $m_n$ and $a^{2n} n^{2n+b} c$ approaches $1$. Compute $\lfloor 2019 abc^2\rfloor$.
[i]Proposed by Vincent Huang[/i]
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Rays $\displaystyle \overrightarrow{OB}$ and $\displaystyle \overrightarrow{DC}$ intersect at $E$, and rays $\displaystyle \overrightarrow{OC}$ and $\displaystyle \overrightarrow{AB}$ intersect at $F$. Suppose that $AE = EC = CF = 4$, and the circumcircle of $ODE$ bisects $\overline{BF}$. Find the area of triangle $ADF$.
[i]Proposed by Howard Halim[/i]
An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common interior point.
(Josef Tkadlec)
Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$.
For how many integers $n\neq1$ does $\left(n-1\right)^3$ divide $n^{2018\left(n-1\right)}-1$?
[i]2018 CCA Math Bonanza Individual Round #12[/i]
Define $\phi^!(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when \[ \sum_{\substack{2 \le n \le 50 \\ \gcd(n,50)=1}} \phi^!(n) \] is divided by $50$.
Find all functions $f : \mathbb R \to \mathbb R$ such that the equality
\[y^2f(x) + x^2f(y) + xy = xyf(x + y) + x^2 + y^2\]
holds for all $x, y \in \Bbb R$, where $\Bbb R$ is the set of real numbers.
Finitely many cities are connected by one-way roads. For any two cities it is possible to come from one of them to the other (with possible transfers), but not necessarily both ways. Prove that there is a city which can be reached from any other city, and that there is a city from which any other city can be reached.
In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then
$$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$
A right triangle $ABC$ with hypotenuse $AB$ is inscribed in a circle. Let $K$ be the midpoint of the arc $BC$ not containing $A, N$ the midpoint of side $AC$, and $M$ a point of intersection of ray $KN$ with the circle. Let $E$ be a point of intersection of tangents to the circle at points $A$ and $C$. Prove that $\angle EMK = 90^\circ$.
Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$,
\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]
a) Show that $I(f) < 3$, for any $f \in \mathcal{F}$.
b) Determine $\sup\{I(f) \mid f \in \mathcal{F}\}$.
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Solve the system of equations:
$
\begin{matrix}
|x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4.
\end{matrix}
$
Consider $f_1=1, f_2=1$ and $f_{n+1}=f_n+f_{n-1}$ for $n\geq 2$. Determine if exists $n\leq 1000001$ such that the last three digits of $f_n$ are zero.
[b]7.1. / 6.5[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers.
[b]7.2[/b] Given a circle $O$ and a square $K$, as well as a line $L$. Construct a segment of given length parallel to $L$ and such that its ends lie on $O$ and $K$ respectively
[b]7.3[/b] The three-digit number $\overline{abc}$ is divisible by $37$. Prove that the sum of the numbers $\overline{bca}$ and $\overline{cab}$ is also divisible by $37$.[b] (typo corrected)[/b]
[b]7.4.[/b] Point $C$ is the midpoint of segment $AB$. On an arbitrary ray drawn from point $C$ and not lying on line $AB$, three consecutive points $P$, $M$ and $Q$ so that $PM=MQ$. Prove that $AP+BQ>2CM$.
[img]https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png[/img]
[b]7.5.[/b] Given $2n+1$ different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].
Find all the positive integers $a,b,c$ such that $3ab= 2c^2$ and $a^3+b^3+c^3$ is the double of a prime number.
Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1.$ Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29,$ for positive integers $m$ and $n$ with $m<n$, find $m+n.$
Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.
Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:
[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]
Let $m,n$ be co-prime integers, such that $m$ is even and $n$ is odd. Prove that the following expression does not depend on the values of $m$ and $n$:
\[ \frac 1{2n} + \sum^{n-1}_{k=1} (-1)^{\left[ \frac{mk}n \right]} \left\{ \frac {mk}n \right\} . \]
[i]Bogdan Enescu[/i]