Found problems: 85335
1978 All Soviet Union Mathematical Olympiad, 266
Prove that for every tetrahedron there exist two planes such that the projection areas on those planes ratio is not less than $\sqrt 2$.
1989 Austrian-Polish Competition, 8
$ABC$ is an acute-angled triangle and $P$ a point inside or on the boundary. The feet of the perpendiculars from $P$ to $BC, CA, AB$ are $A', B', C'$ respectively. Show that if $ABC$ is equilateral, then $\frac{AC'+BA'+CB'}{PA'+PB'+PC'}$ is the same for all positions of $P$, but that for any other triangle it is not.
1993 IMO Shortlist, 1
a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions:
\[ BA = B; \& B^2 = C; \& BC = A;
\]
where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$
b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$
c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is,
\[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.
\]
1949-56 Chisinau City MO, 12
Factor the polynomial $bc (b+c) +ca (c-a)-ab(a + b)$.
2015 Turkey Team Selection Test, 2
There are $2015$ points on a plane and no two distances between them are equal. We call the closest $22$ points to a point its $neighbours$. If $k$ points share the same neighbour, what is the maximum value of $k$?
2007 China Team Selection Test, 2
Let $ x_1, \ldots, x_n$ be $ n>1$ real numbers satisfying $ A\equal{}\left |\sum^n_{i\equal{}1}x_i\right |\not \equal{}0$ and $ B\equal{}\max_{1\leq i<j\leq n}|x_j\minus{}x_i|\not \equal{}0$. Prove that for any $ n$ vectors $ \vec{\alpha_i}$ in the plane, there exists a permutation $ (k_1, \ldots, k_n)$ of the numbers $ (1, \ldots, n)$ such that \[ \left |\sum_{i\equal{}1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A\plus{}B}\max_{1\leq i\leq n}|\alpha_i|.\]
2018 All-Russian Olympiad, 6
$a$ and $b$ are given positive integers. Prove that there are infinitely many positive integers $n$ such that $n^b+1$ doesn't divide $a^n+1$.
2007 Regional Olympiad of Mexico Center Zone, 5
Consider a triangle $ABC$ with $\angle ACB = 2 \angle CAB $ and $\angle ABC> 90 ^ \circ$. Consider the perpendicular on $AC$ that passes through $A$ and intersects $BC$ at $D$, prove that $$\frac {1} {BC} - \frac {2} {DC} = \frac {1} {CA} $$
2018 ELMO Shortlist, 2
Call a number $n$ [i]good[/i] if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers.
(a) Prove that there exist infinitely many sets of $4$ consecutive good numbers.
(b) Find all sets of $5$ consecutive good numbers.
[i]Proposed by Michael Ma[/i]
2018 Caucasus Mathematical Olympiad, 7
Given a positive integer $n>1$. In the cells of an $n\times n$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
2005 May Olympiad, 5
The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this:
[img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img]
The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .
2009 Miklós Schweitzer, 12
Let $ Z_1,\,Z_2\dots,\,Z_n$ be $ d$-dimensional independent random (column) vectors with standard normal distribution, $ n \minus{} 1 > d$. Furthermore let
\[ \overline Z \equal{} \frac {1}{n}\sum_{i \equal{} 1}^n Z_i,\quad S_n \equal{} \frac {1}{n \minus{} 1}\sum_{i \equal{} 1}^n(Z_i \minus{} \overline Z)(Z_i \minus{} \overline Z)^\top\]
be the sample mean and corrected empirical covariance matrix. Consider the standardized samples $ Y_i \equal{} S_n^{ \minus{} 1/2}(Z_i \minus{} \overline Z)$, $ i \equal{} 1,2,\dots,n$. Show that
\[ \frac {E|Y_1 \minus{} Y_2|}{E|Z_1 \minus{} Z_2|} > 1,\]
and that the ratio does not depend on $ d$, only on $ n$.
1982 IMO Longlists, 55
Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.
1986 India National Olympiad, 6
Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.
1988 Romania Team Selection Test, 3
Consider all regular convex and star polygons inscribed in a given circle and having $n$ [i]sides[/i]. We call two such polygons to be equivalent if it is possible to obtain one from the other using a rotation about the center of the circle. How many classes of such polygons exist?
[i]Mircea Becheanu[/i]
2008 Mediterranean Mathematics Olympiad, 3
Let $n$ be a positive integer. Calculate the sum $\sum_{k=1}^n\ \ {\sum_{1\le i_1 < \ldots < i_k\le n}^{}{\frac {2^k}{(i_1 + 1)(i_2 + 1)\ldots (i_k + 1)}}}$
1967 AMC 12/AHSME, 19
The area of a rectangle remains unchanged when it is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2 \frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider. Its area, in square inches, is:
$\textbf{(A)}\ 30\qquad
\textbf{(B)}\ \frac{80}{3}\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ \frac{45}{2}\qquad
\textbf{(E)}\ 20$
2017 CCA Math Bonanza, I5
In the [i]magic square[/i] below, every integer from $1$ to $25$ can be filled in such that the sum in every row, column, and long diagonal is the same. Given that the number in the center square is $18$, what is the sum of the entries in the shaded squares?
[asy]
size(4cm);
for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
}
for (int i = 0; i <= 5; ++i) {
draw((i,0)--(i,5));
}
for (int i = 0; i <= 4; ++i) {
for (int j = 0; j <= 4; ++j) {
if ((i+j)%6 == 1 || (i-j)%6 == 3) {
fill((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle, gray);
}
}
}
label("\Large{18}", (2.5,2.5));
[/asy]
[i]2017 CCA Math Bonanza Individual Round #5[/i]
2018 BMT Spring, 2
A $ 1$ by $ 1$ square $ABCD$ is inscribed in the circle $m$. Circle $n$ has radius $1$ and is centered around $A$. Let $S$ be the set of points inside of $m$ but outside of $n$. What is the area of $S$?
2009 Today's Calculation Of Integral, 501
Find the volume of the uion $ A\cup B\cup C$ of the three subsets $ A,\ B,\ C$ in $ xyz$ space such that:
\[ A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\}\]
\[ B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\}\]
\[ C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}\]
2021 AMC 12/AHSME Spring, 16
In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200.$ $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …, 200, 200, …, 200$$ What is the median of the numbers in this list?
$\textbf{(A)}\ 100.5 \qquad\textbf{(B)}\ 134 \qquad\textbf{(C)}\ 142 \qquad\textbf{(D)}\ 150.5 \qquad\textbf{(E)}\ 167$
1998 All-Russian Olympiad, 2
Two polygons are given on the plane. Assume that the distance between any two vertices of the same polygon is at most 1, and that the distance between any two vertices of different polygons is at least $ 1/\sqrt{2}$. Prove that these two polygons have no common interior points.
By the way, can two sides of a polygon intersect?
2018 Iranian Geometry Olympiad, 1
Two circles $\omega_1,\omega_2$ intersect each other at points $A,B$. Let $PQ$ be a common tangent line of these two circles with $P \in \omega_1$ and $Q \in \omega_2$. An arbitrary point $X$ lies on $\omega_1$. Line $AX$ intersects $ \omega_2$ for the second time at $Y$ . Point $Y'\ne Y$ lies on $\omega_2$ such that $QY = QY'$. Line $Y'B$ intersects $ \omega_1$ for the second time at $X'$. Prove that $PX = PX'$.
Proposed by Morteza Saghafian
1991 Arnold's Trivium, 8
How many maxima, minima, and saddle points does the function $x^4 + y^4 + z^4 + u^4 + v^4$ have on the surface $x+ ... +v = 0$, $x^2+ ... + v^2 = 1$, $x^3 + ... + v^3 = C$?
2002 Manhattan Mathematical Olympiad, 4
Find six points $A_1, A_2, \ldots , A_6$ in the plane, such that for each point $A_i, i = 1, 2, \ldots , 6$ there are exactly three of the remaining five points exactly $1$ cm from $A_i$.