Found problems: 85335
Geometry Mathley 2011-12, 15.2
Let $O$ be the centre of the circumcircle of triangle $ABC$. Point $D$ is on the side $BC$. Let $(K)$ be the circumcircle of $ABD$. $(K)$ meets $AO$ at $E$ that is distinct from $A$.
(a) Prove that $B,K,O,E$ are on the same circle that is called $(L)$.
(b) $(L)$ intersects $AB$ at $F$ distinct $B$. Point $G$ is on $(L)$ such that $EG \parallel OF$. $GK$ meets $AD$ at $S, SO$ meets $BC$ at $T$ . Prove that $O,E, T,C$ are on the same circle.
Trần Quang Hùng
2011 Purple Comet Problems, 17
Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even. For example, $(2, 2, 5, 1)$ and $(3, 1, 4, 6)$ satisfy the conditions.
1952 Putnam, A1
Let \[ f(x) = \sum_{i=0}^{i=n} a_i x^{n - i}\] be a polynomial of degree $n$ with integral coefficients. If $a_0, a_n,$ and $f(1)$ are odd, prove that $f(x) = 0$ has no rational roots.
1971 Miklós Schweitzer, 2
Prove that there exists an ordered set in which every uncountable subset contains an uncountable, well-ordered subset and that cannot be represented as a union of a countable family of well-ordered subsets.
[i]A. Hajnal[/i]
1987 All Soviet Union Mathematical Olympiad, 453
Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$. Prove that the sum of all the numbers is not greater than $1987$.
2021 Estonia Team Selection Test, 1
a) There are $2n$ rays marked in a plane, with $n$ being a natural number. Given that no two marked rays have the same direction and no two marked rays have a common initial point, prove that there exists a line that passes through none of the initial points of the marked rays and intersects with exactly $n$ marked rays.
(b) Would the claim still hold if the assumption that no two marked rays have a common initial point was dropped?
2007 F = Ma, 20
A point-like mass moves horizontally between two walls on a frictionless surface with initial kinetic energy $E$. With every collision with the walls, the mass loses $1/2$ its kinetic energy to thermal energy. How many collisions with the walls are necessary before the speed of the mass is reduced by a factor of $8$?
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $
2019 India PRMO, 6
Let $\overline{abc}$ be a three digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$
2014 Postal Coaching, 4
Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.
1962 AMC 12/AHSME, 39
Two medians of a triangle with unequal sides are $ 3$ inches and $ 6$ inches. Its area is $ 3 \sqrt{15}$ square inches. The length of the third median in inches, is:
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 3 \sqrt{3} \qquad
\textbf{(C)}\ 3 \sqrt{6} \qquad
\textbf{(D)}\ 6 \sqrt{3} \qquad
\textbf{(E)}\ 6 \sqrt{6}$
2018 USA TSTST, 9
Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$ has total area at most $c$.
[i]Linus Hamilton[/i]
2019 IFYM, Sozopol, 6
Does there exist a function $f: \mathbb N \to \mathbb N$ such that for all integers $n \geq 2$,
\[ f(f(n-1)) = f (n+1) - f(n)\, ?\]
1988 Tournament Of Towns, (171) 4
We have a set of weights with masses $1$ gm, $2$ gm, $4$ gm and so on, all values being powers of $2$ . Some of these weights may have equal mass. Some weights were put on both sides of a balance beam, resulting in equilibrium. It is known that on the left hand side all weights were distinct . Prove that on the right hand side there were no fewer weights than on the left hand side.
2004 Italy TST, 2
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
PEN C Problems, 4
Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.
2004 Harvard-MIT Mathematics Tournament, 3
Find \[ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + x^2}-\sqrt[3]{x^3-x^2} \right). \]
1991 Spain Mathematical Olympiad, 2
Given two distinct elements $a,b \in \{-1,0,1\}$, consider the matrix $A$ .
Find a subset $S$ of the set of the rows of $A$, of minimum size, such that every other row of $A$ is a linear combination of the rows in $S$ with integer coefficients.
2016 Taiwan TST Round 1, 2
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
2011 Tuymaada Olympiad, 4
Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.
2023 CCA Math Bonanza, T3
There are exactly 3 distinct 4-digit factors of 3212005. Find their sum.
[i]Team #3[/i]
2022 Macedonian Team Selection Test, Problem 5
Given is an arithmetic progression {$a_n$} of positive integers. Prove that there exist infinitely many $k$, such that $\omega (a_k)$ is even and $\omega (a_{k+1})$ is odd ($\omega (n)$ is the number of distinct prime factors of $n$).
$\textit {Proposed by Viktor Simjanoski and Nikola Velov}$
2004 Cuba MO, 4
Determine all pairs of natural numbers $ (x, y)$ for which it holds that $$x^2 = 4y + 3gcd (x, y).$$
2020 Serbian Mathematical Olympiad, Problem 5
For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions:
$(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$.
$(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$.
Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.
2019 Serbia National Math Olympiad, 6
Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations :
$$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and
$$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$
Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$
2023 MMATHS, 8
Find the number of ordered pairs of integers $(m,n)$ such that $0 \le m,n \le 2023$ and $$m^2 \equiv \sum_{d \mid 2023} n^d \pmod{2024}.$$