Found problems: 85335
2015 Azerbaijan JBMO TST, 1
With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]
2019 CCA Math Bonanza, I2
Square $1$ is drawn with side length $4$. Square $2$ is then drawn inside of Square $1$, with its vertices at the midpoints of the sides of Square $1$. Given Square $n$ for a positive integer $n$, we draw Square $n+1$ with vertices at the midpoints of the sides of Square $n$. For any positive integer $n$, we draw Circle $n$ through the four vertices of Square $n$. What is the area of Circle $7$?
[i]2019 CCA Math Bonanza Individual Round #2[/i]
2005 Georgia Team Selection Test, 11
On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.
2020 Online Math Open Problems, 5
Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$, inclusive, such that, if $$q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}},$$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$.
[i]Proposed by Ankit Bisain[/i]
2011 Romania National Olympiad, 2
Prove that any natural number smaller or equal than the factorial of a natural number $ n $ is the sum of at most $ n $ distinct divisors of the factorial of $ n. $
2022 Bundeswettbewerb Mathematik, 4
For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that:
(a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$.
(b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.
2012 Serbia JBMO TST, 3
Let $a, \overline{bcd}, \overline{aef}, \overline{cfg}, \overline{hci}, \overline{dea}, \overline{ifd}, \overline{jgf}, \overline{bfeg},\ldots$ be an increasing arithmetic progression. Find the $16$th term of this sequence.
2021 Science ON all problems, 1
Consider the prime numbers $p_1,p_2,\dots ,p_{2021}$ such that the sum
$$p_1^4+p_2^4+\dots +p_{2021}^4$$
is divisible by $6060$. Prove that at least $4$ of these prime numbers are less than $2021$.
$\textit{Stefan Bălăucă}$
STEMS 2023 Math Cat A, 4
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:
$\qquad\ (1) \; f(1) = 1.$
$\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$
$\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$
$\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$
Find the sum of all possible values of $f(2023)$.
1972 IMO Longlists, 44
Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.
Revenge EL(S)MO 2024, 5
Inscribe three mutually tangent pink disks of radii $450$, $450$, and $720$ in an uncolored circle $\Omega$ of radius $1200$. In one move, Elmo selects an uncolored region inside $\Omega$ and draws in it the largest possible pink disk. Can Elmo ever draw a disk with a radius that is a perfect square of a rational?
Proposed by [i]Ritwin Narra[/i]
1997 Pre-Preparation Course Examination, 1
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
1995 ITAMO, 1
Determine for which values of $n$ it is possible to tile a square of side $n$ with figures of the type shown in the picture
[asy]
unitsize(0.4 cm);
draw((0,0)--(5,0));
draw((0,1)--(5,1));
draw((1,2)--(4,2));
draw((2,3)--(3,3));
draw((0,0)--(0,1));
draw((1,0)--(1,2));
draw((2,0)--(2,3));
draw((3,0)--(3,3));
draw((4,0)--(4,2));
draw((5,0)--(5,1));
[/asy]
2009 Mid-Michigan MO, 7-9
[b]p1.[/b] Arrange the whole numbers $1$ through $15$ in a row so that the sum of any two adjacent numbers is a perfect square. In how many ways this can be done?
[b]p2.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 - q^2$ is divisible by $24$.
[b]p3.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies.
Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!”
The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!”
A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!”
Then a stripped polyleg started ”None of you has $8$ legs. Only I have $8$ legs!”
Which polyleg has exactly $8$ legs?
[b][b]p4.[/b][/b] There is a small puncture (a point) in the wall (as shown in the figure below to the right). The housekeeper has a small flag of the following form (see the figure left). Show on the figure all the points of the wall where you can hammer in a nail such that if you hang the flag it will close up the puncture.
[img]https://cdn.artofproblemsolving.com/attachments/a/f/8bb55a3fdfb0aff8e62bc4cf20a2d3436f5d90.png[/img]
[b]p5.[/b] Assume $ a, b, c$ are odd integers. Show that the quadratic equation $ax^2 + bx + c = 0$ has no rational solutions.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1
The numbers from 1 to 1996 are written down ------ 12345678910111213.... How many zeros are written?
A. 489
B. 699
C. 796
D. 996
E. None of these
2010 China National Olympiad, 1
Two circles $\Gamma_1$ and $\Gamma_2$ meet at $A$ and $B$. A line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ repsectively. Another line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $E$ and $F$ repsectively. Line $CF$ meets $\Gamma_1$ and $\Gamma_2$ again at $P$ and $Q$ respectively. $M$ and $N$ are midpoints of arc $PB$ and arc $QB$ repsectively. Show that if $CD = EF$, then $C,F,M,N$ are concyclic.
2024 Azerbaijan Senior NMO, 3
In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.
1996 Canada National Olympiad, 2
Find all real solutions to the following system of equations. Carefully justify your answer.
\[ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. \]
1965 AMC 12/AHSME, 19
If $ x^4 \plus{} 4x^3 \plus{} 6px^2 \plus{} 4qx \plus{} r$ is exactly divisible by $ x^3 \plus{} 3x^2 \plus{} 9x \plus{} 3$, the value of $ (p \plus{} q)r$ is:
$ \textbf{(A)}\ \minus{} 18 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 45 \qquad$
2002 Tournament Of Towns, 1
Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.
2010 IFYM, Sozopol, 7
Let $\Delta ABC$ be an isosceles triangle with base $AB$. Point $P\in AB$ is such that $AP=2PB$. Point $Q$ from the segment $CP$ is such that $\angle AQP=\angle ACB$. Prove that $\angle PQB=\frac{1}{2}\angle ACB$.
2014 Argentina National Olympiad Level 2, 1
An [i]operation[/i] on three given non-negative integers consists in increasing two of them by $1$ and decreasing the third by $2$, provided the new numbers are non-negative. The process begins with three non-negative integers that add up to $100$ and are less than $100$. Find the number of distinct triplets that can be obtained by applying the operation. (Triplets that differ only in the order of their members are considered the same).
2018 Spain Mathematical Olympiad, 3
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$.
Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.
2010 CHKMO, 2
There are $ n$ points on the plane, no three of which are collinear. Each pair of points is joined by a red, yellow or green line. For any three points, the sides of the triangle they form consist of exactly two colours. Show that $ n<13$.
2012 National Olympiad First Round, 6
Which one statisfies $n^{29} \equiv 7 \pmod {65}$?
$ \textbf{(A)}\ 37 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 55$