Found problems: 85335
1999 Italy TST, 3
(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \]
(b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]
2019 Final Mathematical Cup, 3
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
2018 Taiwan TST Round 1, 1
Given a triangle $ \triangle{ABC} $ and a point $ O $. $ X $ is a point on the ray $ \overrightarrow{AC} $. Let $ X' $ be a point on the ray $ \overrightarrow{BA} $ so that $ \overline{AX} = \overline{AX_{1}} $ and $ A $ lies in the segment $ \overline{BX_{1}} $. Then, on the ray $ \overrightarrow{BC} $, choose $ X_{2} $ with $ \overline{X_{1}X_{2}} \parallel \overline{OC} $.
Prove that when $ X $ moves on the ray $ \overrightarrow{AC} $, the locus of circumcenter of $ \triangle{BX_{1}X_{2}} $ is a part of a line.
2007 South East Mathematical Olympiad, 1
Determine the number of real number $a$, such that for every $a$, equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:
(a) $x_0$ is an even integer;
(b) $|x_0|<1000$.
2007 Singapore Senior Math Olympiad, 3
In the equilateral triangle $ABC, M, N$ are the midpoints of the sides $AB, AC$, respectively. The line $MN$ intersects the circumcircle of $\vartriangle ABC$ at $K$ and $L$ and the lines $CK$ and $CL$ meet the line $AB$ at $P$ and $Q$, respectively. Prove that $PA^2 \cdot QB = QA^2 \cdot PB$.
2009 Brazil National Olympiad, 2
Let $ q \equal{} 2p\plus{}1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$.
2005 China Team Selection Test, 3
Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
2010 Baltic Way, 4
Find all polynomials $P(x)$ with real coefficients such that
\[(x-2010)P(x+67)=xP(x) \]
for every integer $x$.
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?