Found problems: 191
2013 BAMO, 4
Consider a rectangular array of single digits $d_{i,j}$ with 10 rows and 7 columns, such that $d_{i+1,j}-d_{i,j}$ is always 1 or -9 for all $1 \leq i \leq 9$ and all $1 \leq j \leq 7$, as in the example below. For $1 \leq i \leq 10$, let $m_i$ be the median of $d_{i,1}$, ..., $d_{i,7}$. Determine the least and greatest possible values of the mean of $m_1$, $m_2$, ..., $m_{10}$.
Example:
[img]https://cdn.artofproblemsolving.com/attachments/8/a/b77c0c3aeef14f0f48d02dde830f979eca1afb.png[/img]
1955 Moscow Mathematical Olympiad, 305
$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?
1964 All Russian Mathematical Olympiad, 053
We have to divide a cube onto $k$ non-overlapping tetrahedrons. For what smallest $k$ is it possible?
1988 All Soviet Union Mathematical Olympiad, 484
What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?
1988 All Soviet Union Mathematical Olympiad, 474
In the triangle $ABC$, $\angle C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$.
1984 Tournament Of Towns, (068) T2
A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner?
(Muscovite folklore)
2017 Hanoi Open Mathematics Competitions, 7
Let two positive integers $x, y$ satisfy the condition $44 /( x^2 + y^2)$.
Determine the smallest value of $T = x^3 + y^3$.
2013 Hanoi Open Mathematics Competitions, 11
The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.
1989 All Soviet Union Mathematical Olympiad, 503
Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$.
1980 All Soviet Union Mathematical Olympiad, 300
The $A$ set consists of integers only. Its minimal element is $1$ and its maximal element is $100$. Every element of $A$ except $1$ equals to the sum of two (may be equal) numbers being contained in $A$. What is the least possible number of elements in $A$?
2002 Regional Competition For Advanced Students, 1
Find the smallest natural number $x> 0$ so that all following fractions are simplified
$\frac{3x+9}{8},\frac{3x+10}{9},\frac{3x+11}{10},...,\frac{3x+49}{48}$ , i.e. numerators and denominators are relatively prime.
2005 Spain Mathematical Olympiad, 2
Let $r,s,u,v$ be real numbers. Prove that:
$$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$
1999 Spain Mathematical Olympiad, 6
A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?
2004 Peru MO (ONEM), 4
Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.
2016 Hanoi Open Mathematics Competitions, 15
Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$.
Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.
1988 All Soviet Union Mathematical Olympiad, 480
Find the minimum value of $\frac{xy}{z} + \frac{yz}{x} +\frac{ zx}{y}$ for positive reals $x, y, z$ with $x^2 + y^2 + z^2 = 1$.
2011 Hanoi Open Mathematics Competitions, 8
Find the minimum value of $S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|$.
2017 Hanoi Open Mathematics Competitions, 13
Let $a, b, c$ be the side-lengths of triangle $ABC$ with $a+b+c = 12$.
Determine the smallest value of $M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}$.
2016 India PRMO, 14
Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.
2017 Balkan MO Shortlist, C4
For any set of points $A_1, A_2,...,A_n$ on the plane, one defines $r( A_1, A_2,...,A_n)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \ge 3$, there are indices $i,j,k$ such that $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$
2016 Auckland Mathematical Olympiad, 4
Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers.
1969 Vietnam National Olympiad, 4
Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$.
Find the locus of $H$ and $H'$.
If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value.
Show that this value does not depend on the position of $L$.
Comment on the case $OO'^2 > R^2 + R'^2$.
2016 Romanian Master of Mathematics Shortlist, C1
We start with any finite list of distinct positive integers. We may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.
2013 India PRMO, 7
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?
1965 All Russian Mathematical Olympiad, 056
a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$. What is the minimal possible value of the sum of all products of couples of those numbers.
b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$. What is the minimal possible value of the sum of all products of couples of those numbers.