Found problems: 85335
2023 Simon Marais Mathematical Competition, B1
Find the smallest positive real number $r$ with the following property: For every choice of $2023$ unit vectors $v_1,v_2, \dots ,v_{2023} \in \mathbb{R}^2$, a point $p$ can be found in the plane such that for each subset $S$ of $\{1,2, \dots , 2023\}$, the sum
$$\sum_{i \in S} v_i$$
lies inside the disc $\{x \in \mathbb{R}^2 : ||x-p|| \leq r\}$.
1977 All Soviet Union Mathematical Olympiad, 251
Let us consider one variable polynomials with the senior coefficient equal to one. We shall say that two polynomials $P(x)$ and $Q(x)$ commute, if $P(Q(x))=Q(P(x))$ (i.e. we obtain the same polynomial, having collected the similar terms).
a) For every a find all $Q$ such that the $Q$ degree is not greater than three, and $Q$ commutes with $(x^2 - a)$.
b) Let $P$ be a square polynomial, and $k$ is a natural number. Prove that there is not more than one commuting with $P$ $k$-degree polynomial.
c) Find the $4$-degree and $8$-degree polynomials commuting with the given square polynomial $P$.
d) $R$ and $Q$ commute with the same square polynomial $P$. Prove that $Q$ and $R$ commute.
e) Prove that there exists a sequence $P_2, P_3, ... , P_n, ...$ ($P_k$ is $k$-degree polynomial), such that $P_2(x) = x^2 - 2$, and all the polynomials in this infinite sequence pairwise commute.
2018 PUMaC Individual Finals A, 3
We say that the prime numbers $p_1,\dots,p_n$ construct the graph $G$ if we can assign to each vertex of $G$ a natural number whose prime divisors are among $p_1,\dots,p_n$ and there is an edge between two vertices in $G$ if and only if the numbers assigned to the two vertices have a common divisor greater than $1$. What is the minimal $n$ such that there exist prime numbers $p_1,\dots,p_n$ which construct any graph $G$ with $N$ vertices?
2004 India IMO Training Camp, 4
Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$
1961 AMC 12/AHSME, 20
The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants:
${{ \textbf{(A)}\ \text{I and II} \qquad\textbf{(B)}\ \text{II and III} \qquad\textbf{(C)}\ \text{I and III} \qquad\textbf{(D)}\ \text{III and IV} }\qquad\textbf{(E)}\ \text{I and IV} } $
1969 AMC 12/AHSME, 31
Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,2)--(1,0)--(0,-2)--(-1,0)--cycle);
label("$(0,2)$",(0,2),NE);
label("$(1,0)$",(1,0),SE);
label("$(0,-2)$",(0,-2),SE);
label("$(-1,0)$",(-1,0),SW);
label("$\textbf{(A)}$",(-2,1.5));
[/asy]
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,2)..(1,0)..(0,-2)^^(0,-2)..(-1,0)..(0,2));
label("$(0,2)$",(0,2),NE);
label("$(1,0)$",(1,0),SE);
label("$(0,-2)$",(0,-2),SE);
label("$(-1,0)$",(-1,0),SW);
label("$\textbf{(B)}$",(-2,1.5));
[/asy]
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,2)--(1,0)--(-1,0)--cycle);
label("$(0,2)$",(0,2),NE);
label("$(1,0)$",(1,0),S);
label("$(-1,0)$",(-1,0),S);
label("$\textbf{(C)}$",(-2,1.5));
[/asy]
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,2)..(1/2,3/2)..(1,0)--(-1,0)..(-1/2,3/2)..(0,2));
label("$(0,2)$",(0,2),NE);
label("$(1,0)$",(1,0),S);
label("$(-1,0)$",(-1,0),S);
label("$\textbf{(D)}$",(-2,1.5));
[/asy]
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,1)--(1,0)--(0,-1)--(-1,0)--cycle);
label("$(0,1)$",(0,1),NE);
label("$(1,0)$",(1,0),SE);
label("$(0,-1)$",(0,-1),SE);
label("$(-1,0)$",(-1,0),SW);
label("$\textbf{(E)}$",(-2,1.5));
[/asy]
2022 Auckland Mathematical Olympiad, 7
Points$ D, E, F$ are chosen on the sides $AB$, $BC$, $AC$ of a triangle $ABC$, so that $DE = BE$ and $FE = CE$. Prove that the centre of the circle circumscribed around triangle $ADF$ lies on the bisectrix of angle $DEF$.
2000 Harvard-MIT Mathematics Tournament, 2
If $X=1+x+x^2+x^3+\cdots$ and $Y=1+y+y^2+y^3+\cdots$, what is $1+xy+x^2y^2+x^3y^3+\cdots$ in terms of $X$ and $Y$ only?
2016 Balkan MO Shortlist, A3
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$
[i](Macedonia)[/i]
2010 Stanford Mathematics Tournament, 8
A sphere of radius $1$ is internally tangent to all four faces of a regular tetrahedron. Find the tetrahedron's volume.
2014 Ukraine Team Selection Test, 3
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.
2012 Dutch Mathematical Olympiad, 5
The numbers $1$ to $12$ are arranged in a sequence. The number of ways this can be done equals $12 \times11 \times 10\times ...\times 1$. We impose the condition that in the sequence there should be exactly one number that is smaller than the number directly preceding it.
How many of the $12 \times11 \times 10\times ...\times 1$ sequences satisfy this condition?
2004 Manhattan Mathematical Olympiad, 1
Is there a whole number, so that if we multiply its digits we get $528$?
1955 AMC 12/AHSME, 43
The pairs of values of $ x$ and $ y$ that are the common solutions of the equations $ y\equal{}(x\plus{}1)^2$ and $ xy\plus{}y\equal{}1$ are:
$ \textbf{(A)}\ \text{3 real pairs} \qquad
\textbf{(B)}\ \text{4 real pairs} \qquad
\textbf{(C)}\ \text{4 imaginary pairs} \\
\textbf{(D)}\ \text{2 real and 2 imaginary pairs} \qquad
\textbf{(E)}\ \text{1 real and 2 imaginary pairs}$
2015 Korea Junior Math Olympiad, 3
For all nonnegative integer $i$, there are seven cards with $2^i$ written on it.
How many ways are there to select the cards so that the numbers add up to $n$?
1985 Miklós Schweitzer, 11
Let $\xi (E, \pi, B)\, (\pi\colon E\rightarrow B)$ be a real vector bundle of finite rank, and let
$$\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)$$
be the tangent bundle of $E$, where $V\xi=\mathrm{Ker}\, d\pi$ is the vertical subbundle of $\tau_E$. Let us denote the projection operators corresponding to the splitting $(*)$ by $v$ and $h$. Construct a linear connection $\nabla$ on $V\xi$ such that
$$\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]$$
($X$ and $Y$ are vector fields on $E$, $[.,\, .]$ is the Lie bracket, and all data are of class $\mathcal C^\infty$. [J. Szilasi]
2017 Online Math Open Problems, 13
We define the sets of lattice points $S_0,S_1,\ldots$ as $S_0=\{(0,0)\}$ and $S_k$ consisting of all lattice points that are exactly one unit away from exactly one point in $S_{k-1}$. Determine the number of points in $S_{2017}$.
[i]Proposed by Michael Ren
2014 Contests, Problem 4
Let $\{a_i\}$ be a strictly increasing sequence of positive integers. Define the sequence $\{s_k\}$ as $$s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},$$ where $[a_i,a_{i+1}]$ is the least commun multiple of $a_i$ and $a_{i+1}$.
Show that the sequence $\{s_k\}$ is convergent.
2023 EGMO, 3
Let $k$ be a positive integer. Lexi has a dictionary $\mathbb{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathbb{D}$ when read from top-to-bottom and each row contains a string from $\mathbb{D}$ when read from left-to-right.
What is the smallest integer $m$ such that if $\mathbb{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathbb{D}$?
2014 PUMaC Algebra A, 2
Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?
2003 IMO Shortlist, 3
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles.
[i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]
III Soros Olympiad 1996 - 97 (Russia), 10.10
There are several triangles. From them a new triangle is obtained according to the following rule. The largest side of the new triangle is equal to the sum of the large sides of the data, the middle one is equal to the sum of the middle sides, and the smallest one is the sum of the smaller ones. Prove that if all the angles of these triangles were less than $a$, and $\phi$, where $\phi$ is the largest angle of the resulting triangle, then $\cos \phi \ge 1-\sin (a/2)$.
2005 Georgia Team Selection Test, 9
Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.
2018-2019 Winter SDPC, 2
Call a number [i]precious[/i] if it is the sum of two distinct powers of two. Find all precious numbers $n$ such that $n^2$ is also precious.
1979 IMO Longlists, 32
Let $n, k \ge 1$ be natural numbers. Find the number $A(n, k)$ of solutions in integers of the equation
\[|x_1| + |x_2| +\cdots + |x_k| = n\]