Found problems: 85335
2004 May Olympiad, 5
On a $ 9\times 9$ board, divided into $1\times 1$ squares, pieces of the form
Each piece covers exactly $3$ squares.
(a) Starting from the empty board, what is the maximum number of pieces that can be placed?
(b) Starting from the board with $3$ pieces already placed as shown in the diagram below, what is the maximum number of pieces that can be placed?
[img]https://cdn.artofproblemsolving.com/attachments/d/4/3bd010828accb2d1811d49eb17fa69662ff60d.gif[/img]
2019 Saudi Arabia Pre-TST + Training Tests, 3.1
Let $P(x)$ be a monic polynomial of degree $100$ with $100$ distinct noninteger real roots. Suppose that each of polynomials $P(2x^2 - 4x)$ and $P(4x - 2x^2)$ has exactly $130$ distinct real roots. Prove that there exist non constant polynomials $A(x),B(x)$ such that $A(x)B(x) = P(x)$ and $A(x) = B(x)$ has no root in $(-1.1)$
2012 IMC, 2
Let $n$ be a fixed positive integer. Determine the smallest possible rank of an $n\times n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal.
[i]Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow.[/i]
1946 Putnam, B5
Show that $\lceil (\sqrt{3}+1)^{2n})\rceil$ is divisible by $2^{n+1}.$
1949-56 Chisinau City MO, 16
Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$
2012 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
2000 Vietnam National Olympiad, 2
Two circles $ (O_1)$ and $ (O_2)$ with respective centers $ O_1$, $ O_2$ are given on a plane. Let $ M_1$, $ M_2$ be points on $ (O_1)$, $ (O_2)$ respectively, and let the lines $ O_1M_1$ and $ O_2M_2$ meet at $ Q$. Starting simultaneously from these positions, the points $ M_1$ and $ M_2$ move clockwise on their own circles with the same angular velocity.
(a) Determine the locus of the midpoint of $ M_1M_2$.
(b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.
2004 AMC 12/AHSME, 10
The sum of $ 49$ consecutive integers is $ 7^5$. What is their median?
$ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7^2\qquad \textbf{(C)}\ 7^3\qquad \textbf{(D)}\ 7^4\qquad \textbf{(E)}\ 7^5$
2023 CCA Math Bonanza, L1.3
Let $P$ and $Q$ be two concentric circles, and let $p_1 \dots p_{20}$ be equally spaced points around $P$ and $q_1 \dots q_{23}$ be equally spaced points around $Q$. How many ways are there to connect each $p_i$ to a distinct $q_j$ with some curve (not necessarily a straight line) so that no two curves cross and no curve crosses either circle?
[i]Lightning 1.3[/i]
1942 Putnam, B3
Given $x=\phi(u,v)$ and $y=\psi(u,v)$, where $ \phi$ and $\psi$ are solutions of the partial differential equation
$$(1) \;\,\;\, \; \frac{ \partial \phi}{\partial u} \frac{\partial \psi}{ \partial v} - \frac{ \partial \phi}{\partial v} \frac{\partial \psi}{ \partial u}=1.$$
By assuming that $x$ and $y$ are the independent variables, show that $(1)$ may be transformed to
$$(2) \;\,\;\, \; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}.$$
Integrate $(2)$ and show how this effects in general the solution of $(1)$. What other solutions does $(1)$ possess?
2023 Peru MO (ONEM), 1
We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$.
a) What is the smallest positive integer that divides exactly two elements of $M$?
b) What is the largest positive integer that divides exactly two elements of $M$?
2011 National Olympiad First Round, 33
What is the largest volume of a sphere which touches to a unit sphere internally and touches externally to a regular tetrahedron whose corners are over the unit sphere?
$\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \frac{2\sqrt2}{\sqrt3} - 1 \right ) \qquad\textbf{(E)}\ \text{None}$
2009 Bosnia and Herzegovina Junior BMO TST, 1
Lengths of sides of triangle $ABC$ are positive integers, and smallest side is equal to $2$. Determine the area of triangle $P$ if $v_c = v_a + v_b$, where $v_a$, $v_b$ and $v_c$ are lengths of altitudes in triangle $ABC$ from vertices $A$, $B$ and $C$, respectively.
2014 HMNT, 7
Let $P$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $P$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB = 20$ and $CD = 14$, compute the area of $ABCD$.
2024-25 IOQM India, 30
Let $ABC$ be a right-angled triangle with $\angle B = 90^{\circ}$. Let the length of the altitude $BD$ be equal to $12$. What is the minimum possible length of $AC$, given that $AC$ and the perimeter of triangle $ABC$ are integers?
2022 Rioplatense Mathematical Olympiad, 6
A sequence of numbers is [i]platense[/i] if the first number is greater than $1$, and $a_{n+1}=\frac{a_n}{p_n}$ which $p_n$ is the least prime divisor of $a_n$, and the sequence ends if $a_n=1$. For instance, the sequences $864, 432,216,108,54,27,9,3,1$ and $2022,1011,337,1$ are both sequence platense. A sequence platense is [i]cuboso[/i] if some term is a perfect cube greater than $1$. For instance, the sequence $864$ is cuboso, because $27=3^3$, and the sequence $2022$ is not cuboso, because there is no perfect cube. Determine the number of sequences cuboso which the initial term is less than $2022$.
2019 Kosovo National Mathematical Olympiad, 1
Let $a,b$ be real numbers grater then $4$. Show that at least one of the trinomials $x^2+ax+b$ or $x^2+bx+a$ has two different real zeros.
2014 Iran MO (2nd Round), 1
Find all positive integers $(m,n)$ such that
\[n^{n^{n}}=m^{m}.\]
OIFMAT III 2013, 5
In an acute triangle $ ABC $ with circumcircle $ \Omega $ and circumcenter $ O $, the circle $ \Gamma $ is drawn, passing through the points $ A $, $ O $ and $ C $ together with its diameter $ OQ $, then the points $ M $ and $ N $ are chosen on the lines $ AQ $ and $ AC $, respectively, in such a way that the quadrilateral $ AMBN $ is a parallelogram.
Prove that the point of intersection of the lines $ MN $ and $ BQ $ lies on the circle $ \Gamma $.
1972 Swedish Mathematical Competition, 1
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution.
2002 Miklós Schweitzer, 6
Let $K\subseteq \mathbb{R}$ be compact. Prove that the following two statements are equivalent to each other.
(a) For each point $x$ of $K$ we can assign an uncountable set $F_x\subseteq \mathbb{R}$ such that
$$\mathrm{dist}(F_x, F_y)\ge |x-y|$$
holds for all $x,y\in K$;
(b) $K$ is of measure zero.
2016 Iranian Geometry Olympiad, 5
Let $ABCD$ be a convex quadrilateral with these properties: $\angle ADC = 135^o$ and $\angle ADB - \angle ABD = 2\angle DAB = 4\angle CBD$.
If $BC = \sqrt2 CD$ , prove that $AB = BC + AD$.
by Mahdi Etesami Fard
2002 VJIMC, Problem 2
A ring $R$ (not necessarily commutative) contains at least one non-zero zero divisor and the number of zero divisors is finite. Prove that $R$ is finite.
2024 Harvard-MIT Mathematics Tournament, 3
Compute the number of even positive integers $n \le 2024$ such that $1, 2, \ldots, n$ can be split into $\tfrac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of $3.$
TNO 2008 Senior, 11
Each face of a cube is painted with a different color. How many distinct cubes can be created this way? (*Observation: The ways to color the cube are $6!$, since each time a color is used on one face, there is one fewer available for the others. However, this does not determine $6!$ different cubes, since colorings that differ only by rotation should be considered the same.*)