Found problems: 85335
2011 NIMO Summer Contest, 11
How many ordered pairs of positive integers $(m, n)$ satisfy the system
\begin{align*}
\gcd (m^3, n^2) & = 2^2 \cdot 3^2,
\\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,
\end{align*}
where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?
2018 China Second Round Olympiad, 1
Let $ a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n,A,B$ are positive reals such that $ a_i\leq b_i,a_i\leq A$ $(i=1,2,\cdots,n)$ and $\frac{b_1 b_2 \cdots b_n}{a_1 a_2 \cdots a_n}\leq \frac{B}{A}.$ Prove that$$\frac{(b_1+1) (b_2+1) \cdots (b_n+1)}{(a_1+1) (a_2+1) \cdots (a_n+1)}\leq \frac{B+1}{A+1}.$$
1992 Poland - First Round, 7
Given are the points $A_0 = (0,0,0), A_1 = (1,0,0), A_2 = (0,1,0), A_3 = (0,0,1)$ in the space. Let $P_{ij} (i,j \in 0,1,2,3)$ be the point determined by the equality: $\overrightarrow{A_0P_{ij}} = \overrightarrow{A_iA_j}$. Find the volume of the smallest convex polyhedron which contains all the points $P_{ij}$.
2017 Brazil Team Selection Test, 3
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
[i]Proposed by Evan Chen, Taiwan[/i]
2019 JBMO Shortlist, A3
Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 11 \}$ with $A \cup B = X$. Let
$P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$.
Find the minimum and maximum possible value of $P_A +P_B$ and find all possible equality
cases.
[i]Proposed by Greece[/i]
2019 Taiwan APMO Preliminary Test, P7
Let positive integer $k$ satisfies $1<k<100$. For the permutation of $1,2,...,100$ be $a_1,a_2,...,a_{100}$, take the minimum $m>k$ such that $a_m$ is at least less than $(k-1)$ numbers of $a_1,a_2,...,a_k$. We know that the number of sequences satisfies $a_m=1$ is $\frac{100!}{4}$. Find the all possible values of $k$.
2009 Czech and Slovak Olympiad III A, 6
Given two fixed points $O$ and $G$ in the plane. Find the locus of the vertices of triangles whose circumcenters and centroids are $O$ and $G$ respectively.
2017 Estonia Team Selection Test, 5
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
2000 France Team Selection Test, 2
A function from the positive integers to the positive integers satisfies these properties
1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$.
2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$.
Prove that $f(2)=2, f(3)=3, f(1999)=1999$.
2002 Moldova National Olympiad, 4
At least two of the nonnegative real numbers $ a_1,a_2,...,a_n$ aer nonzero. Decide whether $ a$ or $ b$ is larger if
$ a\equal{}\sqrt[2002]{a_1^{2002}\plus{}a_2^{2002}\plus{}\ldots\plus{}a_n^{2002}}$
and
$ b\equal{}\sqrt[2003]{a_1^{2003}\plus{}a_2^{2003}\plus{}\ldots\plus{}a_n^{2003} }$
2019 Brazil Undergrad MO, 4
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any $(x, y)$ real numbers we have
$f(xf(y)+f(x))+f(y^2)=f(x)+yf(x+y)$
2016 CCA Math Bonanza, T8
As $a$, $b$ and $c$ range over [i]all[/i] real numbers, let $m$ be the smallest possible value of $$2\left(a+b+c\right)^2+\left(ab-4\right)^2+\left(bc-4\right)^2+\left(ca-4\right)^2$$ and $n$ be the number of ordered triplets $\left(a,b,c\right)$ such that the above quantity is minimized. Compute $m+n$.
[i]2016 CCA Math Bonanza Team #8[/i]
1992 Czech And Slovak Olympiad IIIA, 2
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$
2015 IMO, 5
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2010 Today's Calculation Of Integral, 624
Find the continuous function $f(x)$ such that the following equation holds for any real number $x$.
\[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\]
[i]1977 Keio University entrance exam/Medicine[/i]
1992 National High School Mathematics League, 9
From eight edges and eight diagonal of surfaces of a cube, choose $k$ lines. If any two lines of them are skew lines, then the maximum value of $k$ is________.
1978 Putnam, B6
Let $p$ and $n$ be positive integers. Suppose that the numbers $c_{hk}$ ($h=1,2,\ldots,n$ ; $k=1,2,\ldots,ph$) satisfy $0 \leq c_{hk} \leq 1.$ Prove that
$$ \left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,$$
where each summation is over all admissible ordered pairs $(h,k).$
2000 Kazakhstan National Olympiad, 7
Is there any function $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions:
$1) f(0) = 1$
$2) f(x+f(y)) = f(x+y) + 1$, for all $x,y \to\mathbb{R} $
$3)$ there exist rational, but not integer $x_0$, such $f(x_0)$ is integer
1989 Turkey Team Selection Test, 6
The circle, which is tangent to the circumcircle of isosceles triangle $ABC$ ($AB=AC$), is tangent $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that the midpoint $I$ of the segment $PQ$ is the center of the excircle (which is tangent to $BC$) of the triangle .
2021-IMOC, G5
The incircle of a cyclic quadrilateral $ABCD$ tangents the four sides at $E$, $F$, $G$, $H$ in counterclockwise order. Let $I$ be the incenter and $O$ be the circumcenter of $ABCD$. Show that the line connecting the centers of $\odot(OEG)$ and $\odot(OFH)$ is perpendicular to $OI$.
1990 French Mathematical Olympiad, Problem 3
(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$.
(b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.
1969 Putnam, B5
Let $a_1 <a_2 < \ldots$ be an increasing sequence of positive integers. Let the series
$$\sum_{i=1}^{\infty} \frac{1}{a_i }$$
be convergent. For any real number $x$, let $k(x)$ be the number of the $a_i$ which do not exceed $x$. Show
that $\lim_{x\to \infty} \frac{k(x)}{x}=0.$
1988 Irish Math Olympiad, 4
Problem: A mathematical moron is given the values b; c; A for a triangle ABC and
is required to fi nd a. He does this by using the cosine rule
$ a^2 = b^2 + c^2 - 2bccosA$
and misapplying the low of the logarithm to this to get
$ log a^2 = log b^2 + log c^2 - log(2bc cos A) $
He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms
and gets the correct answer. What can be said about the triangle ABC?
2010 ISI B.Math Entrance Exam, 4
If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that
$\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$
2016 BMT Spring, 5
Positive integers $x, y, z$ satisfy $(x + yi)^2 - 46i = z$. What is $x + y + z$?