Found problems: 85335
2025 SEEMOUS, P3
Let $A\in\mathcal{M}_n(\mathbb{C})$ such that $A^*A^2 = AA^*$. Prove that $A^2=A$. (Here we denote by $A^*$ the conjugate transpose of $A$.)
2003 Turkey MO (2nd round), 2
Let $ABCD$ be a convex quadrilateral and $K,L,M,N$ be points on $[AB],[BC],[CD],[DA]$, respectively. Show that,
\[
\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}
\]
where $s_1=\text{Area}(AKN)$, $s_2=\text{Area}(BKL)$, $s_3=\text{Area}(CLM)$, $s_4=\text{Area}(DMN)$ and $s=\text{Area}(ABCD)$.
1986 IMO Longlists, 58
Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.
2015 AIME Problems, 5
Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$.
2013 Cono Sur Olympiad, 2
In a triangle $ABC$, let $M$ be the midpoint of $BC$ and $I$ the incenter of $ABC$. If $IM$ = $IA$, find the least possible measure of $\angle{AIM}$.
BIMO 2022, 1
Given an acute triangle $ABC$, mark $3$ points $X, Y, Z$ in the interior of the triangle. Let $X_1, X_2, X_3$ be the projections of $X$ to $BC, CA, AB$ respectively, and define the points $Y_i, Z_i$ similarly for $i=1, 2, 3$.
a) Suppose that $X_iY_i<X_iZ_i$ for all $i=1,2,3$, prove that $XY<XZ$.
b) Prove that this is not neccesarily true, if triangle $ABC$ is allowed to be obtuse.
[i]Proposed by Ivan Chan Kai Chin[/i]
2016 HMNT, 22-24
22. Let the function $f : \mathbb{Z} \to \mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$, f satisfies
$$f(x) + f(y) = f(x + 1) + f(y - 1)$$
If $f(2016) = 6102$ and $f(6102) = 2016$, what is $f(1)$?
23. Let $d$ be a randomly chosen divisor of $2016$. Find the expected value of
$$\frac{d^2}{d^2 + 2016}$$
24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$?
2021 AMC 12/AHSME Spring, 19
Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10 and the probability of rolling a sum of 12 is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?
$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 19 \qquad\textbf{(E)}\ 20$
2008 ITest, 52
A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers such that $c/d$ is the length of a side of the square, find the value of $c+d$.
2009 CIIM, Problem 1
Prove that for any positive integer $n$ the number $\left(\frac{3+\sqrt{17}}{2}\right)^n+\left(\frac{3-\sqrt{17}}{2}\right)^n $ is an odd integer.
2017 IFYM, Sozopol, 6
The sequence $a_1,a_2…$ , is defined by the equations $a_1=1$ and $a_n=n.a_{[n/2]}$ for $n>1$. Prove that $a_n>n^2$ for $n>11$.
2016 Switzerland Team Selection Test, Problem 11
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
1995 Romania Team Selection Test, 3
Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and that $f(0)$ is a square-free number. Prove that
the polynomial $g(x) = f(x^3)$ is also irreducible
III Soros Olympiad 1996 - 97 (Russia), 11.3
Prove that the equation x^3- x- 3 = 0 has a unique root. Which is greater, the root of this equation or $\sqrt[5]{13}$? (Use of a calculator is prohibited.)
2007 Harvard-MIT Mathematics Tournament, 27
Find the number of $7$-tuples $(n_1,\ldots,n_7)$ of integers such that \[\sum_{i=1}^7 n_i^6=96957.\]
2013 HMNT, 6
Find the number of positive integer divisors of $12! $ that leave a remainder of $1$ when divided by $3$.
2016 Saudi Arabia GMO TST, 3
Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions:
$x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$
2005 France Team Selection Test, 3
In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:
- Any 3 participants speak a common language.
- No language is spoken more that by the half of the participants.
What is the least value of $n$?
2020 Princeton University Math Competition, 12
Given a sequence $a_0, a_1, a_2, ... , a_n$, let its [i]arithmetic approximant[/i] be the arithmetic sequence $b_0, b_1, ... , b_n$ that minimizes the quantity $\sum_{i=0}^{n}(b_i -a_i)^2$, and denote this quantity the sequence’s anti-arithmeticity. Denote the number of integer sequences whose arithmetic approximant is the sequence $4$, $8$, $12$, $16$ and whose anti-arithmeticity is at most $20$.
Kyiv City MO 1984-93 - geometry, 1984.7.3
On the extension of the largest side $AC$ of the triangle $ABC$ set aside the segment $CM$ such that $CM = BC$. Prove that the angle $ABM$ is obtuse or right.
2006 Kyiv Mathematical Festival, 4
See all the problems from 5-th Kyiv math festival
[url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.
2006 Sharygin Geometry Olympiad, 26
Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.
2023 May Olympiad, 2
We say that a four-digit number $\overline{abcd}$ is [i]slippery [/i] if the number $a^4+b^3+c^2+d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ slippery, since $2^4 + 0^3 + 2 ^2 + 3 = 23$. How many slippery numbers are there?
1970 IMO Longlists, 4
Solve the system of equations for variables $x,y$, where $\{a,b\}\in\mathbb{R}$ are constants and $a\neq 0$.
\[x^2 + xy = a^2 + ab\] \[y^2 + xy = a^2 - ab\]
2008 Baltic Way, 14
Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?