Found problems: 85335
2011 Puerto Rico Team Selection Test, 1
A set of ten two-digit numbers is given. Prove that one can always choose two disjoint subsets of this set such that the sum of their elements is the same.
Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )
2011 China Girls Math Olympiad, 4
A tennis tournament has $n>2$ players and any two players play one game against each other (ties are not allowed). After the game these players can be arranged in a circle, such that for any three players $A,B,C$, if $A,B$ are adjacent on the circle, then at least one of $A,B$ won against $C$. Find all possible values for $n$.
2021 MOAA, 11
Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$?
[asy]
size(10cm);
draw((0,0)--(26,0)--(26,10)--(0,10)--cycle);
draw((1,0)--(25,10));
draw(circle((0,5),5));
draw(circle((26,5),5));
dot((1,0));
dot((25,10));
label("$E$",(1,0),SE);
label("$F$",(25,10),NW);
label("$A$", (0,0), SW);
label("$B$", (0,10), NW);
label("$C$", (26,10), NE);
label("$D$", (26,0), SE);
dot((0,0));
dot((0,10));
dot((26,0));
dot((26,10));
[/asy]
[i]Proposed by Nathan Xiong[/i]
1989 AMC 12/AHSME, 5
Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is
[asy]
real xscl = 1.2;
int[] x = {0,1,2,4,5},y={0,1,3,4,5};
for(int a:x){
for(int b:y) {
dot((a*xscl,b));
}
}
for(int a:x) {
pair prev = (a,y[0]);
for(int i = 1;i<y.length;++i) {
pair p = (a,y[i]);
pen pen = linewidth(.7);
if(y[i]-prev.y!=1){
pen+=dotted;
}
draw((xscl*prev.x,prev.y)--(xscl*p.x,p.y),pen);
prev = p;
}
}for(int a:y) {
pair prev = (x[0],a);
for(int i = 1;i<x.length;++i) {
pair p = (x[i],a);
pen pen = linewidth(.7);
if(x[i]-prev.x!=1){
pen+=dotted;
}
draw((xscl*prev.x,prev.y)--(p.x*xscl,p.y),pen);
prev = p;
}
}
path lblx = (0,-.7)--(5*xscl,-.7);
draw(lblx);
label("$10$",lblx);
path lbly = (5*xscl+.7,0)--(5*xscl+.7,5);
draw(lbly);
label("$20$",lbly);[/asy]
$\text{(A)} \ 30 \qquad \text{(B)} \ 200 \qquad \text{(C)} \ 410 \qquad \text{(D)} \ 420 \qquad \text{(E)} \ 430$
2013 Turkey MO (2nd round), 2
Find the maximum value of $M$ for which for all positive real numbers $a, b, c$ we have
\[ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) \]
Fractal Edition 2, P1
The positive integers $a$, $b$, $c$ are such that $\frac{a+b}{b+c}$ is the square of a rational number, and $ab+bc+ca$ is a prime number. Find all possible values of $\frac{a+b}{b+c}$.
1991 Arnold's Trivium, 87
Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.
1966 IMO Longlists, 56
In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.
2020 Iran Team Selection Test, 5
Given $k \in \mathbb{Z}$ prove that there exist infinite pairs of distinct natural numbers such that
\begin{align*}
n+s(2n)=m+s(2m) \\
kn+s(n^2)=km+s(m^2).
\end{align*}
($s(n)$ denotes the sum of digits of $n$.)
[i]Proposed by Mohammadamin Sharifi[/i]
2014 Poland - Second Round, 1.
Let $x, y$ be positive integers such that $\frac{x^2}{y}+\frac{y^2}{x}$ is an integer. Prove that $y|x^2$.
2009 Jozsef Wildt International Math Competition, W. 16
Prove that $$\sum \limits_{k=1}^n \frac{1}{d(k)}>\sqrt{n+1}-1$$ For every $n\geq 1$, $d(n)$ is the number of divisors of $n$
2000 Tournament Of Towns, 1
Determine all real numbers that satisfy the equation $$(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0$$
(RM Kuznec)
1984 Swedish Mathematical Competition, 1
Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.
Russian TST 2020, P2
Octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle $\Omega$ with center $O$. It is known that $A_1A_2\|A_5A_6$, $A_3A_4\|A_7A_8$ and $A_2A_3\|A_5A_8$. The circle $\omega_{12}$ passes through $A_1$, $A_2$ and touches $A_1A_6$; circle $\omega_{34}$ passes through $A_3$, $A_4$ and touches $A_3A_8$; the circle $\omega_{56}$ passes through $A_5$, $A_6$ and touches $A_5A_2$; the circle $\omega_{78}$ passes through $A_7$, $A_8$ and touches $A_7A_4$. The common external tangent to $\omega_{12}$ and $\omega_{34}$ cross the line passing through ${A_1A_6}\cap{A_3A_8}$ and ${A_5A_2}\cap{A_7A_4}$ at the point $X$. Prove that one of the common tangents to $\omega_{56}$ and $\omega_{78}$ passes through $X$.
2008 AMC 10, 25
Michael walks at the rate of $ 5$ feet per second on a long straight path. Trash pails are located every $ 200$ feet along the path. A garbage truck travels at $ 10$ feet per second in the same direction as Michael and stops for $ 30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$
2022 VN Math Olympiad For High School Students, Problem 8
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$.
Prove that: $k(m)$ is even for all $m>2.$
1967 IMO Shortlist, 2
The equation
\[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\]
is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$
2018 Harvard-MIT Mathematics Tournament, 3
A $4\times 4$ window is made out of $16$ square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 1
Suppose $A\in{M_2(\mathbb{C})}$ is not a scalar matrix. Let $S=\{B\in{M_2(\mathbb{C})}|\ AB=BA\}$. If $X,\ Y\in{S}$, then prove that $XY=YX$.
2009 ISI B.Math Entrance Exam, 1
Let $x,y,z$ be non-zero real numbers. Suppose $\alpha, \beta, \gamma$ are complex numbers such that $|\alpha|=|\beta|=|\gamma|=1$. If $x+y+z=0=\alpha x+\beta y+\gamma z$, then prove that $\alpha =\beta =\gamma$.
2003 Junior Tuymaada Olympiad, 8
A few people came to the party. Prove that they can be placed in two rooms so that each of them has in their own room an even number of acquaintances. (One of the rooms can be left empty.)
2009 Argentina Iberoamerican TST, 3
Let $ ABC$ be an isosceles triangle with $ AC \equal{} BC.$ Its incircle touches $ AB$ in $ D$ and $ BC$ in $ E.$ A line distinct of $ AE$ goes through $ A$ and intersects the incircle in $ F$ and $ G.$ Line $ AB$ intersects line $ EF$ and $ EG$ in $ K$ and $ L,$ respectively. Prove that $ DK \equal{} DL.$
2004 Abels Math Contest (Norwegian MO), 4
Among the $n$ inhabitants of an island, where $n$ is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends.
(a) Show that the chief’s order can be achieved by using $n^2/4$ different types of stones.
(b) Prove that this is not necessarily true with less than $n^2/4$ types.
2017 ISI Entrance Examination, 7
Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$:
(a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and
(b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.
2008 ITest, 90
For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.