Found problems: 85335
2001 IberoAmerican, 3
Let $S$ be a set of $n$ elements and $S_1,\ S_2,\dots,\ S_k$ are subsets of $S$ ($k\geq2$), such that every one of them has at least $r$ elements.
Show that there exists $i$ and $j$, with $1\leq{i}<j\leq{k}$, such that the number of common elements of $S_i$ and $S_j$ is greater or equal to: $r-\frac{nk}{4(k-1)}$
1972 Yugoslav Team Selection Test, Problem 4
Determine the largest integer $k(n)$ with the following properties: There exist $k(n)$ different subsets of a given set with $n$ elements such that each two of them have a non-empty intersection.
2023 Assara - South Russian Girl's MO, 7
Given an increasing sequence of different natural numbers $a_1 < a_2 < a_3 < ... < a_n$ such that for any two distinct numbers in this sequence their sum is not divisible by $10$. It is known that $a_n = 2023$.
a) Can $n$ be greater than $800$?
b) What is the largest possible value of $n$?
c) For the value $n$ found in question b), find the number of such sequences with $a_n = 2023$.
1997 Hungary-Israel Binational, 1
Is there an integer $ N$ such that $ \left(\sqrt{1997}\minus{}\sqrt{1996}\right)^{1998}\equal{}\sqrt{N}\minus{}\sqrt{N\minus{}1}$?
2001 AMC 10, 20
A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length $ 2000$. What is the length of each side of the octagon?
$ \textbf{(A)}\ \frac{1}{3}(2000) \qquad
\textbf{(B)}\ 2000(\sqrt2\minus{}1) \qquad
\textbf{(C)}\ 2000(2\minus{}\sqrt2)$
$ \textbf{(D)}\ 1000 \qquad
\textbf{(E)}\ 1000\sqrt2$
2019 NMTC Junior, 6
Find all positive integer triples $(x, y, z) $ that satisfy the equation $$x^4+y^4+z^4=2x^2y^2+2y^2z^2+2z^2x^2-63.$$
2017 India IMO Training Camp, 3
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
1999 India National Olympiad, 6
For which positive integer values of $n$ can the set $\{ 1, 2, 3, \ldots, 4n \}$ be split into $n$ disjoint $4$-element subsets $\{ a,b,c,d \}$ such that in each of these sets $a = \dfrac{b +c +d} {3}$.
1981 Romania Team Selection Tests, 4.
Determine the function $f:\mathbb{R}\to\mathbb{R}$ such that $\forall x\in\mathbb{R}$ \[f(x)+f(\lfloor x\rfloor)f(\{x\})=x,\] and draw its graph. Find all $k\in\mathbb{R}$ for which the equation $f(x)+mx+k=0$ has solutions for any $m\in\mathbb{R}$.
[i]V. Preda and P. Hamburg[/i]
2020 MBMT, 34
Let a set $S$ of $n$ points be called [i]cool[/i] if:
[list]
[*] All points lie in a plane
[*] No three points are collinear
[*] There exists a triangle with three distinct vertices in $S$ such that the triangle contains another point in $S$ strictly inside it
[/list]
Define $g(S)$ for a cool set $S$ to be the sum of the number of points strictly inside each triangle with three distinct vertices in $S$. Let $f(n)$ be the minimal possible value of $g(S)$ across all cool sets of size $n$. Find
\[ f(4) + \dots + f(2020) \pmod{1000}\]
[i]Proposed by Timothy Qian[/i]
2015 Paraguay Juniors, 5
Camila creates a pattern to write the following numbers:
$2, 4$
$5, 7, 9, 11$
$12, 14, 16, 18, 20, 22$
$23, 25, 27, 29, 31, 33, 35, 37$
$…$
Following the same pattern, what is the sum of the numbers in the tenth row?
2011 NIMO Problems, 9
The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$.
[i]Proposed by Eugene Chen
[/i]
2011 Hanoi Open Mathematics Competitions, 10
Two bisectors BD and CE of the triangle ABC intersect at O. Suppose that BD.CE = 2BO.OC. Denote by H the point in BC such that OH perpendicular BC. Prove that AB.AC = 2HB.HC.
2009 Indonesia TST, 4
Given triangle $ ABC$. Let the tangent lines of the circumcircle of $ AB$ at $ B$ and $ C$ meet at $ A_0$. Define $ B_0$ and $ C_0$ similarly.
a) Prove that $ AA_0,BB_0,CC_0$ are concurrent.
b) Let $ K$ be the point of concurrency. Prove that $ KG\parallel BC$ if and only if $ 2a^2\equal{}b^2\plus{}c^2$.
2013 Hitotsubashi University Entrance Examination, 1
Find all pairs $(p,\ q)$ of positive integers such that $3p^3-p^2q-pq^2+3q^3=2013.$
2012 India National Olympiad, 3
Define a sequence $<f_0 (x), f_1 (x), f_2 (x), \dots>$ of functions by $$f_0 (x) = 1$$ $$f_1(x)=x$$ $$(f_n(x))^2 - 1 = f_{n+1}(x) f_{n-1}(x)$$ for $n \ge 1$. Prove that each $f_n (x)$ is a polynomial with integer coefficients.
2018 May Olympiad, 3
Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.
2011 Princeton University Math Competition, B4
Let $f$ be an invertible function defined on the complex numbers such that \[z^2 = f(z + f(iz + f(-z + f(-iz + f(z + \ldots)))))\]
for all complex numbers $z$. Suppose $z_0 \neq 0$ satisfies $f(z_0) = z_0$. Find $1/z_0$.
(Note: an invertible function is one that has an inverse).
1994 Bundeswettbewerb Mathematik, 2
Let $k$ be an integer and define a sequence $a_0 , a_1 ,a_2 ,\ldots$ by
$$ a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.$$
Prove that $a_{n+1} a_n +1$ divides $a_{n+1}^{2} +a_{n}^{2}$ for all $n$.
2007 May Olympiad, 5
You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o$ ,$ \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)
2014-2015 SDML (High School), 9
What is the smallest number of queens that can be placed on an $8\times8$ chess board so that every square is either occupied or can be reached in one move? (A queen can be moved any number of unoccupied squares in a straight line vertically, horizontally, or diagonally.)
$\text{(A) }4\qquad\text{(B) }5\qquad\text{(C) }6\qquad\text{(D) }7\qquad\text{(E) }8$
2021 Math Prize for Girls Problems, 18
Let $N$ be the set of square-free positive integers less than or equal to 50. (A [i]square-free[/i] number is an integer that is not divisible by a perfect square bigger than 1.) How many 3-element subsets $S$ of $N$ are there such that the greatest common divisor of all 3 numbers in $S$ is 1, but no pair of numbers in $S$ is relatively prime?
2010 Ukraine Team Selection Test, 12
Is there a positive integer $n$ for which the following holds:
for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?
1978 AMC 12/AHSME, 19
A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is
$\textbf{(A) }.05\qquad\textbf{(B) }.065\qquad\textbf{(C) }.08\qquad\textbf{(D) }.09\qquad \textbf{(E) }.1$
1993 National High School Mathematics League, 6
$m,n$ are non-zero-real numbers, $z\in\mathbb{C}$. Then, the figure of equations $|z+n\text{i}|+|z-m\text{i}|=n$ and $|z+n\text{i}|-|z-m\text{i}|=-m$ in complex plane is ($F_1,F_2$ are focal points)
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMS84L2RkYWZjM2JmNTc0N2RmYjJlMGUwMGFmMWRkY2RkZTA4NTljZTUwLnBuZw==&rn=MTI0NTI0NTQucG5n[/img]