Found problems: 85335
2023 Israel TST, P2
For each positive integer $n$, define $A(n)$ to be the sum of its divisors, and $B(n)$ to be the sum of products of pairs of its divisors. For example,
\[A(10)=1+2+5+10=18\]
\[B(10)=1\cdot 2+1\cdot 5+1\cdot 10+2\cdot 5+2\cdot 10+5\cdot 10=97\]
Find all positive integers $n$ for which $A(n)$ divides $B(n)$.
2023 Iranian Geometry Olympiad, 2
Let ${I}$ be the incenter of $\triangle {ABC}$ and ${BX}$, ${CY}$ are its two angle bisectors. ${M}$ is the midpoint of arc $\overset{\frown}{BAC}$. It is known that $MXIY$ are concyclic. Prove that the area of quadrilateral $MBIC$ is equal to that of pentagon $BXIYC$.
[i]Proposed by Dominik Burek - Poland[/i]
2017 Yasinsky Geometry Olympiad, 3
Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct using compass and ruler without divisions, a tangent to given circle arc at point $A$ ?
[img]https://1.bp.blogspot.com/-7oQBNJGLsVw/W6dYm4Xw7bI/AAAAAAAAJH8/sJ-rgAQZkW0kvlPOPwYiGjnOXGQZuDnRgCK4BGAYYCw/s1600/Yasinsky%2B2017%2BVIII-IX%2Bp3.png[/img]
2019 Tournament Of Towns, 2
Given a convex pentagon $ABCDE$ such that $AE$ is parallel to $CD$ and $AB=BC$. Angle bisectors of angles $A$ and $C$ intersect at $K$. Prove that $BK$ and $AE$ are parallel.
1989 Greece National Olympiad, 3
From a point $A$ not on line $\varepsilon$, we drop the perpendicular $AB$ on $\varepsilon$ and three other not perpendicular lines $AC$, $AD$,$AE $ which lie on the same semiplane defines by $AB$, such that $(AD )>\frac{1}{2}((AC)+(AE))$. Prove that $(CD )>(DE).$ (Points $B,C,D,,E$ lie on line $\varepsilon$ ) .
2020 Tournament Of Towns, 6
Alice has a deck of $36$ cards, $4$ suits of $9$ cards each. She picks any $18$ cards and gives the rest to Bob. Now each turn Alice picks any of her cards and lays it face-up onto the table, then Bob similarly picks any of his cards and lays it face-up onto the table. If this pair of cards has the same suit or the same value, Bob gains a point. What is the maximum number of points he can guarantee regardless of Alice’s actions?
Mikhail Evdokimov
2010 Iran MO (3rd Round), 6
[b]polyhedral[/b]
we call a $12$-gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points $(0,1)$,$(1,0)$,$(-1,0)$ and $(0,-1)$.
find the faces of the [u]massivest[/u] polyhedral that it's image on every three plane $xy$,$yz$ and $zx$ is a good $12$-gon.
(it's obvios that centers of these three $12$-gons are the origin of coordinates for three dimensions.)
time allowed for this question is 1 hour.
2010 Germany Team Selection Test, 3
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
2013 IFYM, Sozopol, 5
Find all polynomilals $P$ with real coefficients, such that
$(x+1)P(x-1)+(x-1)P(x+1)=2xP(x)$
2024 VJIMC, 1
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function. Prove that
\[\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.\]
1997 Hungary-Israel Binational, 2
The three squares $ACC_{1}A''$, $ABB_{1}'A'$, $BCDE$ are constructed externally on the sides of a triangle $ABC$. Let $P$ be the center of the square $BCDE$. Prove that the lines $A'C$, $A''B$, $PA$ are concurrent.
Durer Math Competition CD Finals - geometry, 2023.C3
$ABC$ is an isosceles triangle. The base $BC$ is $1$ cm long, and legs $AB$ and $AC$ are $2$ cm long. Let the midpoint of $AB$ be $F$, and the midpoint of $AC$ be $G$. Additionally, $k$ is a circle, that is tangent to $AB$ and A$C$, and it’s points of tangency are $F$ and $G$ accordingly. Prove, that the intersection of $CF$ and $BG$ falls on the circle $k$.
2007 Balkan MO Shortlist, G2
Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.
1963 All Russian Mathematical Olympiad, 029
a) Each diagonal of the quadrangle halves its area. Prove that it is a parallelogram.
b) Three main diagonals of the hexagon halve its area. Prove that they intersect in one point.
2021 AMC 12/AHSME Spring, 5
When a student multiplied the number $66$ by the repeating decimal,
$$1. \underline{a} \underline{b} \underline{a} \underline{b} … = 1.\overline{ab},$$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1. \underline{a} \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$- digit integer $\underline{a} \underline{b}$?
$\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$
Gheorghe Țițeica 2024, P2
Let $a,b,c>1$. Solve in $\mathbb{R}$ the equation $\log_{a+b}(a^x+b)=\log_b((b+c)^x-c)$.
[i]Mihai Opincariu[/i]
2010 Slovenia National Olympiad, 5
For what positive integers $n \geq 3$ does there exist a polygon with $n$ vertices (not necessarily convex) with property that each of its sides is parallel to another one of its sides?
2009 Harvard-MIT Mathematics Tournament, 7
Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute
\[
\frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}.
\]
1953 AMC 12/AHSME, 30
A house worth $ \$9000$ is sold by Mr. A to Mr. B at a $ 10\%$ loss. Mr. B sells the house back to Mr. A at a $ 10\%$ gain. The result of the two transactions is:
$ \textbf{(A)}\ \text{Mr. A breaks even} \qquad\textbf{(B)}\ \text{Mr. B gains }\$900 \qquad\textbf{(C)}\ \text{Mr. A loses }\$900\\
\textbf{(D)}\ \text{Mr. A loses }\$810 \qquad\textbf{(E)}\ \text{Mr. B gains }\$1710$
LMT Accuracy Rounds, 2023 S6
Aidan, Boyan, Calvin, Derek, Ephram, and Fanalex are all chamber musicians at a concert. In each performance, any combination of musicians of them can perform for all the others to watch. What is the minimum number of performances necessary to ensure that each musician watches every other musician play?
1992 Tournament Of Towns, (336) 4
Three triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$ are given such that their centres of gravity (intersection points of their medians) lie on a straight line, but no three of the $9$ vertices of the triangles lie on a straight line. Consider the set of $27$ triangles $A_iB_jC_k$ (where $i$, $j$, $k$ take the values $1$, $2$, $3$ independently). Prove that this set of triangles can be divided into two parts of the same total area.
(A. Andjans, Riga)
2004 Turkey MO (2nd round), 1
In a triangle $\triangle ABC$ with$\angle B>\angle C$, the altitude, the angle bisector, and the median from $A$ intersect $BC$ at $H, L$ and $D$, respectively. Show that $\angle HAL=\angle DAL$ if and only if $\angle BAC=90^{\circ}$.
2016 Azerbaijan Team Selection Test, 2
A positive interger $n$ is called [i][u]rising[/u][/i] if its decimal representation $a_ka_{k-1}\cdots a_0$ satisfies the condition $a_k\le a_{k-1}\le\cdots \le a_0$. Polynomial $P$ with real coefficents is called [i][u]interger-valued[/u][/i] if for all integer numbers $n$, $P(n)$ takes interger values. $P(n)$ is called [i][u]rising-valued[/u][/i] if for all [i]rising[/i] numbers $n$, $P(n)$ takes integer values.
Does it necessarily mean that, "every [i]rising-valued[/i] $P$ is also [i]interger-valued[/i] $P$"?
2014 Harvard-MIT Mathematics Tournament, 24
Let $A=\{a_1,a_2,\ldots,a_7\}$ be a set of distinct positive integers such that the mean of the elements of any nonempty subset of $A$ is an integer. Find the smallest possible value of the sum of the elements in $A$.
2018 Mathematical Talent Reward Programme, MCQ: P 5
Let the maximum and minimum value of $f(x)=\cos \left(x^{2018}\right) \sin x$ are $M$ and $m$ respectively where $x \in[-2 \pi, 2 \pi] .$ Then
$$
M+m=
$$
[list=1]
[*] $\frac{1}{2}$
[*] $-\frac{1}{\sqrt{2}}$
[*] $\frac{1}{2018}$
[*] Does not exists
[/list]