Found problems: 85335
2006 Switzerland - Final Round, 4
A circle with circumference 6n units is given and 3n points divide the circumference in n intervals of 1 unit, n intervals of 2 units, and n intervals of 3 units. Prove that there is at least one pair of points that are diametrically opposite to each other.
2012 Online Math Open Problems, 27
$a$ and $b$ are real numbers that satisfy \[a^4+a^2b^2+b^4=900,\] \[a^2+ab+b^2=45.\] Find the value of $2ab.$
[i]Author: Ray Li[/i]
2008 Puerto Rico Team Selection Test, 1
Given a $ 1 \times 25$ rectangle divided into $ 25$ "boxes" ($ 1 \times 1$), is it possible to write integers $ 1$ to $ 25$ so that the sum of any two adjacent "boxes" is a perfect square?
1993 IMO, 1
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$
2012 National Olympiad First Round, 31
$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$.
If $f(6) = 6$, then $f(2012) = ?$
$ \textbf{(A)}\ -2010 \qquad \textbf{(B)}\ -2000 \qquad \textbf{(C)}\ 2000 \qquad \textbf{(D)}\ 2010 \qquad \textbf{(E)}\ 2012$
1988 IMO Shortlist, 10
Let $ N \equal{} \{1,2 \ldots, n\}, n \geq 2.$ A collection $ F \equal{} \{A_1, \ldots, A_t\}$ of subsets $ A_i \subseteq N,$ $ i \equal{} 1, \ldots, t,$ is said to be separating, if for every pair $ \{x,y\} \subseteq N,$ there is a set $ A_i \in F$ so that $ A_i \cap \{x,y\}$ contains just one element. $ F$ is said to be covering, if every element of $ N$ is contained in at least one set $ A_i \in F.$ What is the smallest value $ f(n)$ of $ t,$ so there is a set $ F \equal{} \{A_1, \ldots, A_t\}$ which is simultaneously separating and covering?
2012 Portugal MO, 3
Isabel wants to partition the set $\mathbb{N}$ of the positive integers into $n$ disjoint sets $A_{1}, A_{2}, \ldots, A_{n}$. Suppose that for each $i$ with $1\leq i\leq n$, given any positive integers $r, s\in A_{i}$ with $r\neq s$, we have $r+s\in A_{i}$. If $|A_{j}|=1$ for some $j$, find the greatest positive integer that may belong to $A_{j}$.
1993 Korea - Final Round, 1
Consider a $9 \times 9$ array of white squares. Find the largest $n \in\mathbb N$ with the property: No matter how one chooses $n$ out of 81 white squares and color in black, there always remains a $1 \times 4$ array of white squares (either vertical or horizontal).
2004 Baltic Way, 20
Three fixed circles pass through the points $A$ and $B$. Let $X$ be a variable point on the first circle different from $A$ and $B$. The line $AX$ intersects the other two circles at $Y$ and $Z$ (with $Y$ between $X$ and $Z$). Show that the ratio $\frac{XY}{YZ}$ is independent of the position of $X$.
2016 JBMO Shortlist, 1
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$ .
Find the least positive integer $k$ making the number $k!\cdot S_{2016}$ an integer.
2005 Polish MO Finals, 3
Let be a convex polygon with $n >5$ vertices and area $1$. Prove that there exists a convex hexagon inside the given polygon with area at least $\dfrac{3}{4}$
2021 Purple Comet Problems, 14
Each of the cells of a $7 \times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of hidden edges is less than $3$.
2013 Switzerland - Final Round, 9
Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$
2020/2021 Tournament of Towns, P4
There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer $N{}$ no matter how the cat plays?
[i]Ivan Mitrofanov[/i]
2003 Germany Team Selection Test, 2
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.
1998 Harvard-MIT Mathematics Tournament, 6
In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?
[asy]
size(340);
int i, j;
for(i = 0; i<10; i = i+1) {
for(j = 0; j<5; j = j+1) {
if(10*j + i == 11 || 10*j + i == 12 || 10*j + i == 14 || 10*j + i == 15 || 10*j + i == 18 || 10*j + i == 32 || 10*j + i == 35 || 10*j + i == 38 ) { }
else{ label("$*$", (i,j));}
}}
label("$\leftarrow$"+"Dec. 31", (10.3,0));
label("Jan. 1"+"$\rightarrow$", (-1.3,4));[/asy]
2012 Gulf Math Olympiad, 1
Let $X,\ Y$ and $Z$ be the midpoints of sides $BC,\ CA$, and $AB$ of the triangle $ABC$, respectively. Let $P$ be a point inside the triangle. Prove that the quadrilaterals $AZPY,\ BXPZ$, and $CYPX$ have equal areas if, and only if, $P$ is the centroid of $ABC$.
2019 Jozsef Wildt International Math Competition, W. 69
Denote $\overline{w_a}, \overline{w_b}, \overline{w_c}$ the external angle-bisectors in triangle $ABC$, prove that $$\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}$$
2021 Brazil Undergrad MO, Problem 4
For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$.
Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$
2010 Tournament Of Towns, 1
There are $100$ points on the plane. All $4950$ pairwise distances between two points have been recorded.
$(a)$ A single record has been erased. Is it always possible to restore it using the remaining records?
$(b)$ Suppose no three points are on a line, and $k$ records were erased. What is the maximum value of $k$ such that restoration of all the erased records is always possible?
2024 Canadian Mathematical Olympiad Qualification, 7a
In triangle $ABC$, let $I$ be the incentre. Let $H$ be the orthocentre of triangle $BIC$. Show that $AH$ is parallel to $BC$ if and only if $H$ lies on the circle with diameter $AI$.
2012 Dutch BxMO/EGMO TST, 4
Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$.
Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.
1993 Cono Sur Olympiad, 2
Prove that there exists a succession $a_1, a_2, ... , a_k, ...$, where each $a_i$ is a digit ($a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)$ ) and $a_0=6$, such that, for each positive integrer $n$, the number $x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1}$ verify that $x_n^2-x_n$ is divisible by $10^n$.
2007 Junior Balkan Team Selection Tests - Romania, 4
Find all integer positive numbers $n$ such that:
$n=[a,b]+[b,c]+[c,a]$, where $a,b,c$ are integer positive numbers and $[p,q]$ represents the least common multiple of numbers $p,q$.
2014 Flanders Math Olympiad, 1
(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals.
(b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that
$$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$