Found problems: 85335
Each subset of $97$ out of $1997$ given real numbers has positive sum.
Show that the sum of all the $1997$ numbers is positive.
Let $a$ be the sum of the numbers:
$99 \times 0.9$
$999 \times 0.9$
$9999 \times 0.9$
$\vdots$
$999\cdots 9 \times 0.9$
where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$.
Find the sum of the digits in the number $a$.
$a^3 + b^3 + 3abc \ge\ c^3$ prove that where a,b and c are sides of triangle.
Consider all planes through the center of a $2\times2\times2$ cube that create cross sections that are regular polygons. The sum of the cross sections for each of these planes can be written in the form $a\sqrt b+c$, where $b$ is a square-free positive integer. Find $a+b+c$.
Let $x_1,x_2,\ldots,x_n$ be positive reals. Prove that
\[ \frac 1{1+x_1} + \frac 1{1+x_1+x_2} + \cdots + \frac 1{1+x_1+\cdots + x_n} < \sqrt { \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n}} . \]
[i]Bogdan Enescu[/i]
Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that
(i) $m<2a$;
(ii) $2n|(2am-m^2+n^2)$;
(iii) $n^2-m^2+2mn\leq2a(n-m)$.
For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\]
Determine the maximum and minimum values of $f$.
[b]E[/b]ach of the integers $1,2,...,729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122...$ How many times in this string does the substring $012$ appear?
Define an [i]almost-palindrome[/i] as a string of letters that is not a palindrome but can become a palindrome if one of its letters is changed. For example, $TRUST$ is an almost-palindrome because the $R$ can be changed to an $S$ to produce a palindrome, but $TRIVIAL$ is not an almost-palindrome because it cannot be changed into a palindrome by swapping out only one letter (both the $A$ and the $L$ are out of place). How many almost-palindromes contain fewer than $4$ letters.
Find all functions $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$ such that
$$xf(x)+(f(y))^2+2xf(y)$$
is perfect square for all positive integers $x,y$.
**This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.
Let $\frac{35x-29}{x^2-3x+2}=\frac{N_1}{x-1}+\frac{N_2}{x-2}$ be an identity in $x$. The numerical value of $N_1N_2$ is:
$\text{(A)} \ -246 \qquad \text{(B)} \ -210 \qquad \text{(C)} \ -29 \qquad \text{(D)} \ 210 \qquad \text{(E)} \ 246$
The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.
Given quadrilateral $ABCD$ inscribed into a circle with diagonal $AC$ as diameter. Let $E$ be a point on segment $BC$ s.t. $\sphericalangle DAC=\sphericalangle EAB$. Point $M$ is midpoint of $CE$. Prove that $BM=DM$.
Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$.
Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that
$$
p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1.
$$
[i](I. Gorodnin)[/i]
Let $p$ be a prime number. Positive integers numbers $a$ and $b$ are such $\frac{p}{a}+\frac{p}{b}=1$ and $a+b$ is divisible by $p$. What values can an expression $\frac{a+b}{p}$ take?
[i]Yu.A.Karpenko[/i]
Let $m,n,p$ be three positive integers, and let $m'=\gcd(m,np)$, $n'=\gcd(n,pm)$ and $p'=\gcd(p,mn)$. Prove that the equation $x^m+y^n=z^p$ has solutions in the set of positive integers if and only if the equation $x^{m'}+y^{n'}=z^{p'}$ has solutions in the set of positive integers.
[i]Luminița Popescu[/i]
Two towns, $A$ and $B$, are connected by a straight road, $15$ miles long. Traveling from town $A$ to town $B$, the speed limit changes every $5$ miles: from $25$ to $40$ to $20$ miles per hour (mph). Two cars, one at town $A$ and one at town $B$, start moving toward each other at the same time. They drive exactly the speed limit in each portion of the road. How far from town $A$, in miles, will the two cars meet?
$\textbf{(A) }7.75 \qquad\textbf{(B) }8 \qquad\textbf{(C) }8.25\qquad\textbf{(D) }8.5 \qquad\textbf{(E) }8.75$
Find all integer solutions to \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997}.\]
On a square of a chessboard there is a pawn . Two players take turns to move it to another square, subject to the rule that , at each move the distance moved is strictly greater than that of the previous move. A player loses when unable to make a move on his turn. Who wins if the players always choose the best strategy? (The pawn is always placed in the centre of its square. )
( F . L . Nazarov)
Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones.
Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible.
(a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains.
(b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.
Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two.
[i]Proposed by Morteza Saghafian[/i]
Let $ABCD$ be a square of side length 2, and let $M$ and $N$ be points on the sides $AB$ and $CD$ respectively. The lines $CM$ and $BN$ meet at $P$, while the lines $AN$ and $DM$ meet at $Q$. Prove that $\left| PQ \right|\ge 1$.
Let $k > 1$ be a fixed odd number, and for non-negative integers $n$ let
$$f_n=\sum_{\substack{0\leq i\leq n\\ k\mid n-2i}}\binom{n}{i}.$$
Prove that $f_n$ satisfy the following recursion:
$$f_{n}^2=\sum_{i=0}^{n} \binom{n}{i}f_{i}f_{n-i}.$$
Let \( ABC \) be a triangle and \( D \) the foot of the altitude from \( A \). Let \( M \) be a point such that \( MB = MC \). Let \( E \) and \( F \) be the intersections of the circumcircle of \( BMD \) and \( CMD \) with \( AD \). Let \( G \) and \( H \) be the intersections of \( MB \) and \( MC \) with \( AD \). Prove that \( EG = FH \).
Does there exist an positive integer $n$, so that for any positive integer $m<1002$, there exists an integer $k$ so that \[\displaystyle \frac{m}{1002} < \frac{k}{n} < \frac {m+1}{1003}\] holds? If $n$ does not exist, prove it; if $n$ exists, determine the minimum value of it.
I know this problem was easy, but it still appeared on our TST, and so I posted it here.
A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is
$ \textbf{(A)}\ 4\plus{}\sqrt{185} \qquad
\textbf{(B)}\ 16 \qquad
\textbf{(C)}\ 17 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ \sqrt{32}\plus{}\sqrt{137}$