Found problems: 85335
1996 Mexico National Olympiad, 3
Prove that it is not possible to cover a $6\times 6$ square board with eighteen $2\times 1$ rectangles, in such a way that each of the lines going along the interior gridlines cuts at least one of the rectangles. Show also that it is possible to cover a $6\times 5$ rectangle with fifteen $2\times 1 $ rectangles so that the above condition is fulfilled.
2007 China Team Selection Test, 2
After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$
2011 Pre-Preparation Course Examination, 3
a government has decided to help it's people by giving them $n$ coupons for $n$ fundamental things, but because of being unmanaged, the giving of the coupons to the people is random. in each time that a person goes to the office to get a coupon, the office manager gives him one of the $n$ coupons randomly and with the same probability. It's obvious that in this system a person may get a coupon that he had it before.
suppose that $X_n$ is the random varieble of the first time that a person gets all of the $n$ coupons. show that $\frac{X_n}{n ln(n)}$ in probability converges to $1$.
2020 China Northern MO, BP3
Are there infinitely many positive integers $n$ such that $19|1+2^n+3^n+4^n$? Justify your claim.
2022 CCA Math Bonanza, T10
Evan, Larry, and Alex are drawing whales on the whiteboard. Evan draws 10 whales, Larry draws 15 whales, and Alex draws 20 whales. Michelle then starts randomly erasing whales one by one. The probability that she finishes erasing Larry's whales first can be expressed as $\frac{p}{q}$. Compute $p+q$.
[i]2022 CCA Math Bonanza Team Round #10[/i]
2022 Greece JBMO TST, 3
The real numbers $x,y,z$ are such that $x+y+z=4$ and $0 \le x,y,z \le 2$. Find the minimun value of the expression $$A=\sqrt{2+x}+\sqrt{2+y}+\sqrt{2+z}+\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}$$.
2007 Harvard-MIT Mathematics Tournament, 14
We are given some similar triangles. Their areas are $1^2,3^2,5^2,\cdots,$ and $49^2$. If the smallest triangle has a perimeter of $4$, what is the sum of all the triangles' perimeters?
2010 AMC 10, 9
Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$?
$ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$
1985 Miklós Schweitzer, 12
Let $(\Omega, \mathcal A, P)$ be a probability space, and let $(X_n, \mathcal F_n)$ be an adapted sequence in $(\Omega, \mathcal A, P)$ (that is, for the $\sigma$-algebras $\mathcal F_n$, we have $\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A$, and for all $n$, $X_n$ is an $\mathcal F_n$-measurable and integrable random variable). Assume that
$$\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\,\, (n=2, 3, \ldots )$$
Prove that $\mathrm{sup}_n \mathrm{E}|X_n|<\infty$ implies that $X_n$ converges with probability one as $n\to\infty$. [I. Fazekas]
2016 Estonia Team Selection Test, 9
Let $n$ be a positive integer such that there exists a positive integer that is less than $\sqrt{n}$ and does not divide $n$. Let $(a_1, . . . , a_n)$ be an arbitrary permutation of $1, . . . , n$. Let $a_{i1} < . . . < a_{ik}$ be its maximal increasing subsequence and let $a_{j1} > . . . > a_{jl}$ be its maximal decreasing subsequence.
Prove that tuples $(a_{i1}, . . . , a_{ik})$ and $(a_{j1}, . . . , a_{jl} )$ altogether contain at least one number that does not divide $n$.
2021 Miklós Schweitzer, 6
Let $f$ and $g$ be $2 \pi$-periodic integrable functions such that in some neighborhood of $0$, $g(x) = f(ax)$ with some $a \neq 0$. Prove that the Fourier series of $f$ and $g$ are simultaneously convergent or divergent at $0$.
2004 BAMO, 5
Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions.
1. $f (0) = 2004$.
2. If $x$ is irrational, then $f (x)$ is also irrational.
(Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients.
A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)
2020 March Advanced Contest, 1
In terms of \(a\), \(b\), and a prime \(p\), find an expression which gives the number of \(x \in \{0, 1, \ldots, p-1\}\) such that the remainder of \(ax\) upon division by \(p\) is less than the remainder of \(bx\) upon division by \(p\).
2013 Sharygin Geometry Olympiad, 3
Let $ABC$ be a right-angled triangle ($\angle B = 90^\circ$). The excircle inscribed into the angle $A$ touches the extensions of the sides $AB$, $AC$ at points $A_1, A_2$ respectively; points $C_1, C_2$ are defined similarly. Prove that the perpendiculars from $A, B, C$ to $C_1C_2, A_1C_1, A_1A_2$ respectively, concur.
1952 AMC 12/AHSME, 14
A house and store were sold for $ \$12000$ each. The house was sold at a loss of $ 20\%$ of the cost, and the store at a gain of $ 20\%$ of the cost. The entire transaction resulted in:
$ \textbf{(A)}\ \text{no loss or gain} \qquad\textbf{(B)}\ \text{loss of } \$1000 \qquad\textbf{(C)}\ \text{gain of } \$1000 \qquad\textbf{(D)}\ \text{gain of }\$2000 \qquad\textbf{(E)}\ \text{none of these}$
2010 Today's Calculation Of Integral, 597
In space given a board shaped the equilateral triangle $PQR$ with vertices $P\left(1,\ \frac 12,\ 0\right),\ Q\left(1,-\frac 12,\ 0\right),\ R\left(\frac 14,\ 0,\ \frac{\sqrt{3}}{4}\right)$. When $S$ is revolved about the $z$-axis, find the volume of the solid generated by the whole points through which $S$ passes.
1984 Tokyo University entrance exam/Science
2023 Centroamerican and Caribbean Math Olympiad, 4
A four-digit number $n=\overline{a b c d}$, where $a, b, c$ and $d$ are digits, with $a \neq 0$, is said to be [i]guanaco[/i] if the product $\overline{a b} \times \overline{c d}$ is a positive divisor of $n$. Find all guanaco numbers.
2004 Mediterranean Mathematics Olympiad, 3
Let $a,b,c>0$ and $ab+bc+ca+2abc=1$ then prove that
\[2(a+b+c)+1\geq 32abc\]
2009 Indonesia TST, 4
Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.
2006 Junior Balkan Team Selection Tests - Romania, 2
Consider the integers $a_1, a_2, a_3, a_4, b_1, b_2, b_3, b_4$ with $a_k \ne b_k$ for all $k = 1, 2, 3, 4$. If
$\{a_1, b_1\} + \{a_2, b_2\} = \{a_3, b_3\} + \{a_4, b_4\}$, show that the number $|(a_1 - b_1)(a_2 - b_2)(a_3 - b_3)(a_4 - b_4)|$ is a square.
Note. For any sets $A$ and $B$, we denote $A + B = \{x + y | x \in A, y \in B\}$.
2016 Auckland Mathematical Olympiad, 4
Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers.
2014 Harvard-MIT Mathematics Tournament, 3
[4] Let $ABCDEF$ be a regular hexagon. Let $P$ be the circle inscribed in $\triangle{BDF}$. Find the ratio of the area of circle $P$ to the area of rectangle $ABDE$.
2021 Moldova Team Selection Test, 5
Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.
2006 Stanford Mathematics Tournament, 9
If to the numerator and denominator of the fraction $ \frac{1}{3}$ you add its denominator 3, the fraction will double. Find a fraction which will triple when its denominator is added to its numerator and to its denominator and find one that will quadruple.
2011 Bogdan Stan, 3
Let be a sequence of real numbers $ \left( x_n \right)_{n\ge 1} $ chosen such that the limit of the sequence $ \left(
x_{n+2011}-x_n \right)_{n\ge 1} $ exists. Calculate $ \lim_{n\to\infty } \frac{x_n}{n} . $
[i]Cosmin Nițu[/i]