Found problems: 85335
2005 Cono Sur Olympiad, 3
The monetary unit of a certain country is called Reo, and all the coins circulating are integers values of Reos. In a group of three people, each one has 60 Reos in coins (but we don't know what kind of coins each one has). Each of the three people can pay each other any integer value between 1 and 15 Reos, including, perhaps with change. Show that the three persons together can pay exactly (without change) any integer value between 45 and 135 Reos, inclusive.
2008 Germany Team Selection Test, 3
Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$.
[i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]
2020 Denmark MO - Mohr Contest, 3
Which positive integers satisfy the following three conditions?
a) The number consists of at least two digits.
b) The last digit is not zero.
c) Inserting a zero between the last two digits yields a number divisible by the original number.
2018 PUMaC Geometry A, 4
Triangle $ABC$ has $\angle{A}=90^\circ$, $\angle{C}=30^\circ$, and $AC=12$. Let the circumcircle of this triangle
be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle{CAD}=60^\circ$. Define
points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively.
Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$
intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c}-\sqrt{d})$
for positive integers $a,b,c,d,$ where $a,b$ are relatively prime, and the sum of $a,b,c,d$ is minimal.
Find $a+b+c+d$.
2008 Harvard-MIT Mathematics Tournament, 4
Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $ 1$ Joule of energy to hop one step north or one step south, and $ 1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $ 100$ Joules of energy, and hops till he falls asleep with $ 0$ energy. How many different places could he have gone to sleep?
2005 Taiwan TST Round 1, 1
Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$
My solution was nearly complete...
2002 Tuymaada Olympiad, 4
A rectangular table with 2001 rows and 2002 columns is partitioned into $1\times 2$ rectangles. It is known that any other partition of the table into $1\times 2$ rectangles contains a rectangle belonging to the original partition.
Prove that the original partition contains two successive columns covered by 2001 horizontal rectangles.
[i]Proposed by S. Volchenkov[/i]
1999 Belarusian National Olympiad, 6
Solve in integer:$x^6+x^2y=y^3+2y^2$
2001 Estonia National Olympiad, 3
A circle with center $I$ and radius $r$ is inscribed in a triangle $ABC$ with a right angle at $C$. Rays $AI$ and $CI$ meet the opposite sides at $D$ and $E$ respectively. Prove that $\frac{1}{AE}+\frac{1}{BD}=\frac{1}{r}$
2008 Postal Coaching, 4
Find all functions $f : R \to R$ such that $$f(xf(y))= (1 - y)f(xy) + x^2y^2f(y)$$
for all reals $x, y$.
2020 LIMIT Category 1, 11
In $\triangle ABC$, $\angle A=30^{\circ}$, $BC=13$. Given $2$ circles $\gamma_1, \gamma_2$ ith radius $r_1,r_2$ contain $A$ and touch $BC$ at $B$ and $C$ respectively. Find $r_1r_2$.
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$.