Found problems: 85335
2024 SG Originals, Q4
Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\dots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2\dots,x_n\in \mathbb{R}$. Bob wins if he can pick a vector $(z_1,z_2,\dots,z_n)\in S$ such that $$\sum_{i=1}^n x_iy_i\le \sum_{i=1}^n x_iz_i,$$otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win.
[i]Proposed by DVDthe1st[/i]
2000 Italy TST, 3
Given positive numbers $a_1$ and $b_1$, consider the sequences defined by
\[a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)\]
Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.
2008 Harvard-MIT Mathematics Tournament, 31
Let $ \mathcal{C}$ be the hyperbola $ y^2 \minus{} x^2 \equal{} 1$. Given a point $ P_0$ on the $ x$-axis, we construct a sequence of points $ (P_n)$ on the $ x$-axis in the following manner: let $ \ell_n$ be the line with slope $ 1$ passing passing through $ P_n$, then $ P_{n\plus{}1}$ is the orthogonal projection of the point of intersection of $ \ell_n$ and $ \mathcal C$ onto the $ x$-axis. (If $ P_n \equal{} 0$, then the sequence simply terminates.)
Let $ N$ be the number of starting positions $ P_0$ on the $ x$-axis such that $ P_0 \equal{} P_{2008}$. Determine the remainder of $ N$ when divided by $ 2008$.
2023 BMT, 6
In triangle $\vartriangle ABC$, let $M$ be the midpoint of $\overline{AC}$. Extend $\overline{BM}$ such that it intersects the circumcircle of $\vartriangle ABC$ at a point $X$ not equal to $B$. Let $O$ be the center of the circumcircle of $\vartriangle ABC$. Given that $BM = 4MX$ and $\angle ABC = 45^o$, compute $\sin (\angle BOX)$.
2024 AMC 12/AHSME, 11
Let $x_{n} = \sin^2(n^\circ)$. What is the mean of $x_{1}, x_{2}, x_{3}, \cdots, x_{90}$?
$
\textbf{(A) }\frac{11}{45} \qquad
\textbf{(B) }\frac{22}{45} \qquad
\textbf{(C) }\frac{89}{180} \qquad
\textbf{(D) }\frac{1}{2} \qquad
\textbf{(E) }\frac{91}{180} \qquad
$
2021 AMC 12/AHSME Spring, 12
All the roots of polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers. What is the value of $B$?
$\textbf{(A)}\ -88 \qquad\textbf{(B)}\ -80 \qquad\textbf{(C)}\ -64\qquad\textbf{(D)}\ -41 \qquad\textbf{(E)}\ -40$
1981 IMO Shortlist, 5
A cube is assembled with $27$ white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.
Russian TST 2017, P2
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
2002 Kurschak Competition, 1
We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$, $O$, $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$, then there must be another vertex on this circle as well.
2014 Hanoi Open Mathematics Competitions, 8
Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.
2022 MIG, 22
How many ways are there to color each of the $8$ cells below red or blue such that no two blue cells are adjacent?
[asy]
size(3cm);
draw((0,0)--(4,0)--(4,1)--(0,1)--(0,0));
draw((1,-1)--(1,2)--(3,2)--(3,-1)--(1,-1));
draw((2,-1)--(2,2));
[/asy]
$\textbf{(A) }48\qquad\textbf{(B) }50\qquad\textbf{(C) }52\qquad\textbf{(D) }54\qquad\textbf{(E) }56$
1993 Putnam, B1
What is the smallest integer $n > 0$ such that for any integer m in the range $1, 2, 3, ... , 1992$ we can always find an integral multiple of $\frac{1}{n}$ in the open interval $(\frac{m}{1993}, \frac{m+1}{1994})$?
2012 Pan African, 2
Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(x^2 - y^2) = (x+y)(f(x) - f(y))$ for all real numbers $x$ and $y$.
2020 Princeton University Math Competition, A7
Suppose that $p$ is the unique monic polynomial of minimal degree such that its coefficients are rational numbers and one of its roots is $\sin \frac{2\pi}{7} + \cos \frac{4\pi}{7}$. If $p(1) = \frac{a}{b}$, where $a, b$ are relatively prime integers, find $|a + b|$.
1994 ITAMO, 1
Show that there exists an integer $N$ such that for all $n \ge N$ a square can be partitioned into $n$ smaller squares.
2022 China Second Round A2, 3
$S=\{1,2,...,N\}$. $A_1,A_2,A_3,A_4\subseteq S$, each having cardinality $500$. $\forall x,y\in S$, $\exists i\in\{1,2,3,4\}$, $x,y\in A_i$. Determine the maximal value of $N$.
2018 Online Math Open Problems, 7
A quadrilateral and a pentagon (both not self-intersecting) intersect each other at $N$ distinct points, where $N$ is a positive integer. What is the maximal possible value of $N$?
[i]Proposed by James Lin
1998 National Olympiad First Round, 17
In triangle $ ABC$, internal bisector of angle $ A$ intersects with $ BC$ at $ D$. Let $ E$ be a point on $ \left[CB\right.$ such that $ \left|DE\right|\equal{}\left|DB\right|\plus{}\left|BE\right|$. The circle through $ A$, $ D$, $ E$ intersects $ AB$ at $ F$, again. If $ \left|BE\right|\equal{}\left|AC\right|\equal{}7$, $ \left|AD\right|\equal{}2\sqrt{7}$ and $ \left|AB\right|\equal{}5$, then $ \left|BF\right|$ is
$\textbf{(A)}\ \frac {7\sqrt {5} }{5} \qquad\textbf{(B)}\ \sqrt {7} \qquad\textbf{(C)}\ 2\sqrt {2} \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \sqrt {10}$
2000 Irish Math Olympiad, 4
The sequence $ a_1<a_2<...<a_M$ of real numbers is called a weak arithmetic progression of length $ M$ if there exists an arithmetic progression $ x_0,x_1,...,x_M$ such that:
$ x_0 \le a_1<x_1 \le a_2<x_2 \le ... \le a_M<x_M.$
$ (a)$ Prove that if $ a_1<a_2<a_3$ then $ (a_1,a_2,a_3)$ is a weak arithmetic progression.
$ (b)$ Prove that any subset of $ \{ 0,1,2,...,999 \}$ with at least $ 730$ elements contains a weak arithmetic progression of length $ 10$.
1978 Germany Team Selection Test, 5
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:
(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$
(ii) some plane contains exactly three points from $E.$
2007 Stars of Mathematics, 2
Let be a structure formed by $ n\ge 4 $ points in space, four by four noncoplanar, and two by two connected by a wire. If we cut the $ n-1 $ wires that connect a point to the others, the remaining point is said to be [i]isolated.[/i] The structure is said to be [i]disconnected[/i] if there are at least two points for which there isn´t a chain of wires connecting them. So, initially, it´s not disconnected.
$ \text{(1)} $ Prove that, by cutting a number smaller or equal with $ n-2, $ the structure won´t become disconnected.
$ \text{(2)} $ Determine the minimum number of wires that needs to be cut so that the remaining structure is disconnected, yet every point, not isloated.
2018 AMC 10, 22
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$?
$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$
1976 Czech and Slovak Olympiad III A, 4
Determine all solutions of the linear system of equations
\begin{align*}
&x_1& &-x_2& &-x_3& &-\cdots& &-x_n& &= 2a, \\
-&x_1& &+3x_2& &-x_3& &-\cdots& &-x_n& &= 4a, \\
-&x_1& &-x_2& &+7x_3& &-\cdots& &-x_n& &= 8a, \\
&&&&&&&&&&&\vdots \\
-&x_1& &-x_2& &-x_3& &-\cdots& &+\left(2^n-1\right)x_n& &= 2^na,
\end{align*}
with unknowns $x_1,\ldots,x_n$ and a real parameter $a.$
2013 IFYM, Sozopol, 7
Let $T$ be a set of natural numbers, each of which is greater than 1. A subset $S$ of $T$ is called “good”, if for each $t\in T$ there exists $s\in S$, for which $gcd(t,s)>1$. Prove that the number of "good" subsets of $T$ is odd.
2011 International Zhautykov Olympiad, 3
Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)