Found problems: 85335
2005 Austrian-Polish Competition, 6
Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$
\[f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}\]
2001 AMC 10, 16
The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum?
$ \textbf{(A)} \ 5 \qquad \textbf{(B)} \ 20 \qquad \textbf{(C)} \ 25 \qquad \textbf{(D)} \ 30 \qquad \textbf{(E)} \ 36$
2010 AIME Problems, 5
Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.
2020 Durer Math Competition Finals, 7
Santa Claus plays a guessing game with Marvin before giving him his present. He hides the present behind one of $100$ doors, numbered from $1$ to $100$. Marvin can point at a door, and then Santa Claus will reply with one of the following words:
$\bullet$ "hot" if the present lies behind the guessed door,
$\bullet$ "warm" if the guess is not exact but the number of the guessed door differs from that of the present’s door by at most $5$,
$\bullet$ "cold" if the numbers of the two doors differ by more than $5$.
At least how many such guesses does Marvin need, so that he can be certain about where his present is?
Marvin does not necessarily need to make a "hot" guess, just to know the correct door with $100\%$ certainty.
2011 NIMO Summer Contest, 14
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
[i]Proposed by Eugene Chen
[/i]
2021 Peru PAGMO TST, P3
Find all the quaterns $(x,y,z,w)$ of real numbers (not necessarily distinct) that solve the following system of equations:
$$x+y=z^2+w^2+6zw$$
$$x+z=y^2+w^2+6yw$$
$$x+w=y^2+z^2+6yz$$
$$y+z=x^2+w^2+6xw$$
$$y+w=x^2+z^2+6xz$$
$$z+w=x^2+y^2+6xy$$
2009 QEDMO 6th, 4
Let $a$ and $b$ be two real numbers and let $n$ be a nonnegative integer. Then prove that
$$\sum_{k=0}^{n} {n \choose k} (a + k)^k (b - k)^{n-k} = \sum_{k=0}^{n} \frac{n!}{t!} (a + b)^t $$
2019 Middle European Mathematical Olympiad, 1
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds
$$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$
[i]Proposed by Patrik Bak, Slovakia[/i]
2016 AIME Problems, 14
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P$ and $Q$ is $d$. Find $d$.
2023 Indonesia TST, C
There are $2023$ distinct points on a plane, which are coloured in white or red, such that for each white point, there are exactly two red points whose distance is $2023$ to that white point. Find the minimum number of red points.
1991 Polish MO Finals, 1
Prove or disprove that there exist two tetrahedra $T_1$ and $T_2$ such that:
(i) the volume of $T_1$ is greater than that of $T_2$;
(ii) the area of any face of $T_1$ does not exceed the area of any face of $T_2$.
2009 Baltic Way, 15
A unit square is cut into $m$ quadrilaterals $Q_1,\ldots ,Q_m$. For each $i=1,\ldots ,m$ let $S_i$ be the sum of the squares of the four sides of $Q_i$. Prove that
\[S_1+\ldots +S_m\ge 4\]
2012 Benelux, 2
Find all quadruples $(a,b,c,d)$ of positive real numbers such that $abcd=1,a^{2012}+2012b=2012c+d^{2012}$ and $2012a+b^{2012}=c^{2012}+2012d$.
2012 May Olympiad, 4
Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.
2019 Peru IMO TST, 1
In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that:
[LIST]
[*] There are no two numbers written in the first row that are equal to each other.[/*]
[*] The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*]
[*] The two numbers written in each column are different and they add up to a rational number.[/*]
[/LIST]
Determine the maximum quantity of irrational numbers that can be in the chessboard.
PEN I Problems, 18
Do there exist irrational numbers $a, b>1$ and $\lfloor a^{m}\rfloor \not=\lfloor b^{n}\rfloor $ for any positive integers $m$ and $n$?
2000 AIME Problems, 12
The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$
2011 Indonesia TST, 4
A prime number $p$ is a [b]moderate[/b] number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \]
If $q$ is the smallest [b]moderate[/b] number, then determine the smallest prime $r$ which is not moderate and $q < r$.
1966 IMO Shortlist, 7
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
2003 Austrian-Polish Competition, 4
A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.
2002 National High School Mathematics League, 9
Points $P_1,P_2,P_3,P_4$ are vertexes of a regular triangular pyramid, and $P_5,P_6,P_7,P_8,P_9,P_{10}$ midpoints of edges. The number of groups $(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)$ that $P_1,P_i,P_j,P_k$ are coplane is________.
2024 LMT Fall, 21
Let $ABC$ be a triangle such that $AB=2$, $BC=3$, and $AC=4$. A circle passing through $A$ intersects $AB$ at $D$, $AC$ at $E$, and $BC$ at $M$ and $N$ such that $BM=MN=NC$. Find $DE$.
2001 Tuymaada Olympiad, 6
On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$. Prove that $PQ \le \frac{1}{2} AB$.
2010 Princeton University Math Competition, 8
Let $N$ be the number of (positive) divisors of $2010^{2010}$ ending in the digit $2$. What is the remainder when $N$ is divided by 2010?
2013 Kazakhstan National Olympiad, 1
Find all triples of positive integer $(m,n,k)$ such that $ k^m|m^n-1$ and $ k^n|n^m-1$