Found problems: 85335
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.)
Let $ABC$ be an acute triangle such that $BA = BC$. On the sides $BA$ and $BC$ points $D$ and $E$ are chosen respectively, such that $DE$ and $AC$ are parallel. Let $H$ be the orthocenter of the triangle $DBE$ and $M$ be the midpoint of $AE$. If $\angle HMC = 90^o$, determine the measure of angle $\angle ABC$.
For how many integer value of $ m$ does the lines $ 13x\plus{}11y \equal{} 700$ and $ y \equal{} mx\minus{}1$ intersect in a point with integer valued coordinats?
A. None
B. 1
C. 2
D. 3
E. Infinitely many
On a straight line, three distinct points $ M $, $ D $, $ H $ are given. Construct a right-angled triangle for which $ M $ is the midpoint of the hypotenuse, $ D $ is the point of intersection of the bisector of the right angle with the hypotenuse, and $ H $ is the foot of the altitude to the hypotenuse.
Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies
$$|a^{2021}-N|<2^{-1000}.$$
[i] (Vlad Robu) [/i]
Let $ABC$ be an acute triangle (with $AB \ne AC$) and $M$ be the midpoint of $BC$. The circle of diameter $AM$ cuts $AC$ at $N$ and $BC$ again at $H$. A point $K$ is taken on $AC$ (between $A$ and $N$) such that $CN = NK$. Segments $AH$ and $BK$ intersect at $L$. The circle that passes through $A$,$K$ and $L$ cuts $AB$ at $P$. Prove that $C$,$L$ and $P$ are collinear.
10. Let $n \ge 5$ be an odd number and let $r$ be an integer such that $1\le r \le (n-1)/2$. IN a sports tournament, $n$ players take part in a series of contests. In each contest, $2r+1$ players participate, and the scores obtained by the players are the numbers
$$-r, -(r-1),\cdots, -1, 0, 1 \cdots, r-1, r$$
in some order. Each possible subset of $2r+1$ players takes part together in exactly one contest. let the final score of player $i$ be $S_i$, for each $i=1, 2,\cdots,n$. Define $N$ to be the smallest difference between the final scores of two players, i.e.,
$$N = \min_{i<j}|S_i - S_j|.$$
Determine, with proof, the maximum possible value of $N$.
The diagonals of a trapezoid $ABCD$ with $AD \parallel BC$ intersect at point $P$. Point $Q$ lies between the parallel lines $AD$ and $BC$ such that the line $CD$ separates points $P$ and $Q$, and $\angle AQD=\angle CQB$. Prove that $\angle BQP = \angle DAQ$.
We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ .
(V. Prasolov)
$101$ numbers are written on a blackboard: $1^2, 2^2, 3^2, \cdots, 101^2$. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.
Let Akbar and Birbal together have $n$ marbles, where $n > 0$.
Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.”
Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.”
What is the minimum possible value of $n$ for which the above statements are true?
Two distinct positive integers are called [i]close [/i] if their greatest common divisor equals their difference. Show that for any $n$, there exists a set $S$ of $n$ elements such that any two elements of $S$ are close.
Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.
Find all positive integer solutions $ (a,b,c,n)$ of the equation: $ 2^n\equal{}a!\plus{}b!\plus{}c!$.
Let $D$ and $E$ be foots of altitudes from $A$ and $B$ of triangle $ABC$, $F$ be intersection point of angle bisector from $C$ with side $AB$, and $O$, $I$ and $H$ be circumcenter, center of inscribed circle and orthocenter of triangle $ABC$, respectively. If $\frac{CF}{AD}+ \frac{CF}{BE}=2$, prove that $OI = IH$.
There are $12$ seats at a round table in a restaurant. A group of five women and seven men arrives at the table. How many ways are there for choosing the sitting order, provided that every woman ought to be surrounded by two men and two orders are regarded as different, if at least one person has a different neighbour on one's right side.
From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.
Find all $ f:\mathbb{Z}\rightarrow\mathbb{Z} $ such that
\[ f(2m+f(m)+f(m)f(n))=nf(m)+m \] $ \forall m,n\in\mathbb{Z} $
Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$.
(1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$.
(2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$.
(3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then
find the smallest $n$ such that $a_n=0$.
Let $n\ge 2$ be an integer. Elwyn is given an $n\times n$ table filled with real numbers (each cell of the table contains exactly one number). We define a [i]rook set[/i] as a set of $n$ cells of the table situated in $n$ distinct rows as well as in n distinct columns. Assume that, for every rook set, the sum of $n$ numbers in the cells forming the set is nonnegative.\\
\\ By a move, Elwyn chooses a row, a column, and a real number $a,$ and then he adds $a$ to each number in the chosen row, and subtracts $a$ from each number in the chosen column (thus, the number at the intersection of the chosen row and column does not change). Prove that Elwyn can perform a sequence of moves so that all numbers in the table become nonnegative.
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$
[i](Macedonia)[/i]
The number halfway between $ \frac {1}{8}$ and $ \displaystyle \frac {1}{10}$ is
$ \textbf{(A)}\ \frac {1}{80} \qquad \textbf{(B)}\ \frac {1}{40} \qquad \textbf{(C)}\ \frac {1}{18} \qquad \textbf{(D)}\ \frac {1}{9} \qquad \textbf{(E)}\ \frac {9}{80}$
Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.
Let $I\subseteq \mathbb{R}$ be an interval and $f:I\rightarrow\mathbb{R}$ a function such that:
\[|f(x)-f(y)|\le |x-y|,\quad\text{for all}\ x,y\in I. \]
Show that $f$ is monotonic on $I$ if and only if, for any $x,y\in I$, either $f(x)\le f\left(\frac{x+y}{2}\right)\le f(y)$ or $f(y)\le f\left(\frac{x+y}{2}\right)\le f(x)$.
What is the smallest positive integer with $24$ positive divisors?