Found problems: 85335
1953 AMC 12/AHSME, 9
The number of ounces of water needed to reduce $ 9$ ounces of shaving lotion containing $ 50\%$ alcohol to a lotion containing $ 30\%$ alcohol is:
$ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 7$
2016 Miklós Schweitzer, 7
Show that the unit sphere bundle of the $r$-fold direct sum of the tautological (universal) complex line bundle over the space $\mathbb{C}P^{\infty}$ is homotopically equivalent to $\mathbb{C}P^{r-1}$.
1994 AMC 12/AHSME, 27
A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white?
$ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3} $
2025 AMC 8, 15
Kei draws a $6\times 6$ grid. He colors $13$ of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let $m$ and $M$ the least and greatest possible number of gold-on-gold pairs, respectively. What is $m + M?$
$\textbf{(A) } 12 \qquad\textbf{(B) }14 \qquad\textbf{(C) }16\qquad\textbf{(D) }18 \qquad\textbf{(E) }20$\\
2022 Rioplatense Mathematical Olympiad, 5
Let $ABCDEFGHI$ be a regular polygon with $9$ sides and the vertices are written in the counterclockwise and let $ABJKLM$ be a regular polygon with $6$ sides and the vertices are written in the clockwise. Prove that $\angle HMG=\angle KEL$.
Note: The polygon $ABJKLM$ is inside of $ABCDEFGHI$.
2014 PUMaC Algebra B, 1
Evaluate $\tfrac1{\sqrt1+\sqrt2}+\tfrac1{\sqrt2+\sqrt3}+\cdots+\tfrac1{\sqrt{1368}+\sqrt{1369}}$.
2005 China Team Selection Test, 2
Let $a$, $b$, $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$. Prove that
\[\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3 \]
2009 Jozsef Wildt International Math Competition, W. 6
Prove that$$p (n)= 2+ \left (p (1) + \cdots + p\left ( \left [\frac {n}{2} \right ] + \chi_1 (n)\right ) + \left (p'_2(n) + \cdots + p' _{ \left [\frac {n}{2} \right ] - 1}(n)\right )\right )$$for every $n \in \mathbb {N}$ with $n>2$ where $\chi $ denotes the principal character Dirichlet modulo 2, i.e.$$ \chi _1 (n) = \begin{cases} 1 & \text{if } (n,2)=1 \\ 0 &\text{if } (n,2)>1 \end{cases} $$with $p (n) $ we denote number of possible partitions of $n $ and $p' _m(n) $ we denote the number of partitions of $n$ in exactly $m$ sumands.
2010 Benelux, 4
Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and
\[a^3 + b^3 = p^n\mbox{.}\]
[i](2nd Benelux Mathematical Olympiad 2010, Problem 4)[/i]
PEN D Problems, 23
Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list]
2021 ELMO Problems, 3
Each cell of a $100\times 100$ grid is colored with one of $101$ colors. A cell is [i]diverse[/i] if, among the $199$ cells in its row or column, every color appears at least once. Determine the maximum possible number of diverse cells.
2001 Korea - Final Round, 3
Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that
\[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\]
and find all cases of equality.
2002 All-Russian Olympiad Regional Round, 8.6
Each side of the convex quadrilateral was continued into both sides and on all eight extensions set aside equal segments. It turned out that the resulting $8$ points are the outer ends of the construction the given segments are different and lie on the same circle. Prove that the original quadrilateral is a square.
2008 China Western Mathematical Olympiad, 3
For a given positive integer $n$, find the greatest positive integer $k$, such that there exist three sets of $k$ non-negative distinct integers, $A=\{x_1,x_2,\cdots,x_k\}, B=\{y_1,y_2,\cdots,y_k\}$ and $C=\{z_1,z_2,\cdots,z_k\}$ with $ x_j\plus{}y_j\plus{}z_j\equal{}n$ for any $ 1\leq j\leq k$.
[size=85][color=#0000FF][Moderator edit: LaTeXified][/color][/size]
2012 AMC 8, 24
A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?
[asy]
size(0,50);
draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle);
dot((-1,1));
dot((-2,2));
dot((-3,1));
dot((-2,0));
draw((1,0){up}..{left}(0,1));
dot((1,0));
dot((0,1));
draw((0,1){right}..{up}(1,2));
dot((1,2));
draw((1,2){down}..{right}(2,1));
dot((2,1));
draw((2,1){left}..{down}(1,0));
[/asy]
$\textbf{(A)}\hspace{.05in}\dfrac{4-\pi}\pi \qquad \textbf{(B)}\hspace{.05in}\dfrac1\pi \qquad \textbf{(C)}\hspace{.05in}\dfrac{\sqrt2}{\pi} \qquad \textbf{(D)}\hspace{.05in}\dfrac{\pi-1}\pi \qquad \textbf{(E)}\hspace{.05in}\dfrac3\pi $
2021 Romania National Olympiad, 1
Let $ABC$ be an acute-angled triangle with the circumcenter $O$. Let $D$ be the foot of the altitude from $A$. If $OD\parallel AB$, show that $\sin 2B = \cot C$.
[i]Mădălin Mitrofan[/i]
2024 Indonesia TST, A
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.
Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
2005 Germany Team Selection Test, 2
For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.
1953 Moscow Mathematical Olympiad, 257
Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2 < 10^{-9}$.
2014 Thailand TSTST, 2
Let $a, b, c$ be positive real numbers. Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}\geq\sqrt{\frac{2ab}{3a+b+2c}}+\sqrt{\frac{2bc}{3b+c+2a}}+\sqrt{\frac{2ca}{3c+a+2b}}.$$
2010 China Western Mathematical Olympiad, 1
Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that
[b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$;
[b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.
1990 Tournament Of Towns, (252) 6
We call a collection of weights (each weighing an integer value) basic if their total weight equals $200$ and each object of integer weight not greater than $200$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equalc value are chosen to balance against a particular object, if in fact there is a choice.)
(a) Find an example of a basic collection other than the collection of $200$ weights each of value $1$.
(b) How many different basic collections are there?
(D. Fomin, Leningrad)
2014 Math Hour Olympiad, 8-10.7
If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$.
Show that the sequence
$\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$
contains infinitely many odd numbers.
2016 Harvard-MIT Mathematics Tournament, 6
The numbers $1, 2\ldots11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
2021 MOAA, 23
Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andrew Wen[/i]