Found problems: 85335
1991 IMO Shortlist, 23
Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that
[i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$
[i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$
1984 IMO Longlists, 24
(a) Decide whether the fields of the $8 \times 8$ chessboard can be numbered by the numbers $1, 2, \dots , 64$ in such a way that the sum of the four numbers in each of its parts of one of the forms
[list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28446[/img][/list]
is divisible by four.
(b) Solve the analogous problem for
[list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28447[/img][/list]
2022/2023 Tournament of Towns, P6
The midpoints of all heights of a certain tetrahedron lie on its inscribed sphere. Is this tetrahedron necessarily regular then?
2011 Purple Comet Problems, 29
Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$.
2016 Hanoi Open Mathematics Competitions, 7
Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points?
2024 AMC 12/AHSME, 16
A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?
$
\textbf{(A) }5 \qquad
\textbf{(B) }6 \qquad
\textbf{(C) }7 \qquad
\textbf{(D) }8 \qquad
\textbf{(E) }9 \qquad
$
2014 Cuba MO, 8
Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.
2015 AMC 10, 17
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
[asy]
import three; size(2inch);
currentprojection=orthographic(4,2,2);
draw((0,0,0)--(0,0,3),dashed);
draw((0,0,0)--(0,4,0),dashed);
draw((0,0,0)--(5,0,0),dashed);
draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3));
draw((0,4,3)--(5,4,3)--(5,4,0));
label("3",(5,0,3)--(5,0,0),W);
label("4",(5,0,0)--(5,4,0),S);
label("5",(5,4,0)--(0,4,0),SE);
[/asy]
$\textbf{(A) } \dfrac{75}{12}
\qquad\textbf{(B) } 10
\qquad\textbf{(C) } 12
\qquad\textbf{(D) } 10\sqrt2
\qquad\textbf{(E) } 15
$
2021 Iran MO (3rd Round), 1
Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that
$$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$
2024 Caucasus Mathematical Olympiad, 4
Yasha writes in the cells of the table $99 \times 99$ all positive integers from 1 to $99^2$ (each number once). Grisha looks at the table and selects several cells, among which there are no two cells sharing a common side, and then sums up the numbers in all selected cells. Find the largest sum Grisha can guarantee to achieve.
2003 Purple Comet Problems, 5
Let $a$, $b$, and $c$ be nonzero real numbers such that $a + \frac{1}{b} = 5$, $b + \frac{1}{c} = 12$, and $c + \frac{1}{a} = 13$. Find $abc + \frac{1}{abc}$.
2012 Gheorghe Vranceanu, 2
$ G $ is the centroid of $ ABC. $ The incircle of $ ABC $ touches $ BC,CA,AB $ at $ D,E,F, $ respectively. Show that $ ABC $ is equilateral if and only if $ BC\cdot\overrightarrow{GD}+ AC\cdot\overrightarrow{GE} +AB\cdot\overrightarrow{GF} =0. $
[i]Marian Ursărescu[/i]
Kvant 2019, M2568
15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible
to have equal numbers of apricots in all the boxes after $k$ moves.
2004 Unirea, 3
Hello,
I've been trying to solve this for a while now, but no success! I mean, I have an expression for this but not a closed one. I derived something in terms of Tchebychev Polynomials : cos(nx) = P_n(cos(x)). Any help is appreciated.
Compute the following primitive:
\[ \int \frac{x\sin\left(2004 x\right)}{\tan x}\ dx\]
Russian TST 2015, P4
Let $p \geq 5$ be a prime number. Prove that there exists a positive integer $a < p-1$ such that neither of $a^{p-1}-1$ and $(a+1)^{p-1}-1$ is divisible by $p^{2}$ .
ICMC 3, 1
An [I]automorphism[/i] of a group \(\left(G,*\right)\) is a bijective function \(f:G\to G\) satisfying \(f(x*y)=f(x)*f(y)\) for all \(x,y\in G\).
Find a group \((G,*)\) with fewer than \((201.6)^2=40642.56\) unique elements and exactly \(2016^2\) unique automorphisms.
[i]Proposed by the ICMC Problem Committee[/i]
1929 Eotvos Mathematical Competition, 1
In how many ways can the sum of 100 fillér be made up with coins of denominations l, 2, 10, 20 and 50 fillér?
2008 Mongolia Team Selection Test, 1
Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$
2005 VJIMC, Problem 3
Let $f:[0,1]\times[0,1]\to\mathbb R$ be a continuous function. Find the limit
$$\lim_{n\to\infty}\left(\frac{(2n+1)!}{(n!)^2}\right)^2\int^1_0\int^1_0(xy(1-x)(1-y))^nf(x,y)\text dx\text dy.$$
2000 AMC 8, 15
Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$?
[asy]
pair A,B,C,D,EE,F,G;
A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3));
EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2);
draw(A--B--C--cycle);
draw(D--EE--A);
draw(EE--F--G);
label("$A$",A,S);
label("$B$",B,SW);
label("$C$",C,N);
label("$D$",D,NE);
label("$E$",EE,NE);
label("$F$",F,SE);
label("$G$",G,SE);
[/asy]
$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21$
1997 AMC 12/AHSME, 17
A line $ x \equal{} k$ intersects the graph of $ y \equal{} \log_5{x}$ and the graph of $ y \equal{} \log_5{(x \plus{} 4)}$. The distance between the points of intersection is $ 0.5$. Given that $ k \equal{} a \plus{} \sqrt{b}$, where $ a$ and $ b$ are integers, what is $ a \plus{} b$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 8\qquad
\textbf{(D)}\ 9\qquad
\textbf{(E)}\ 10$
2003 Estonia National Olympiad, 2
Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.
1994 Chile National Olympiad, 6
On a sheet of transparent paper, draw a quadrilateral with Chinese ink, which is illuminated with a lamp. Show that it is always possible to locate the sheet in such a way that the shadow projected on the desk is a parallelogram.
2022 CCA Math Bonanza, L1.1
Given
$$a = bc$$
$$b = ca$$
$$c = a + b$$
$$c > a$$
Evaluate $a+b+c$.
[i]2022 CCA Math Bonanza Lightning Round 1.1[/i]
2005 Korea National Olympiad, 2
For triangle $ABC$, $P$ and $Q$ satisfy $\angle BPA + \angle AQC=90^{\circ}$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \neq N$, however if $A$ is the only intersection $A=N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.