Found problems: 85335
2005 MOP Homework, 5
Find all integer solutions to $y^2(x^2+y^2-2xy-x-y)=(x+y)^2(x-y)$.
2004 Purple Comet Problems, 11
Find the sum of all integers $x$ satisfying $1 + 8x \le 358 - 2x \le 6x + 94$.
1987 IMO Longlists, 32
Solve the equation $28^x = 19^y +87^z$, where $x, y, z$ are integers.
1998 South africa National Olympiad, 5
Prove that \[ \gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)} \] is a prime if $n$ is a power of a prime, and 1 otherwise.
2018 Korea National Olympiad, 4
Find all real values of $K$ which satisfies the following.
Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$.
(i). $0 < a_n < n^K$.
(ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$.
Then, there exists a positive integer $N$ such that for all integers $n>N$, $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$
2006 QEDMO 2nd, 12
Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.
1966 All Russian Mathematical Olympiad, 078
Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.
2014 Albania Round 2, 3
In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$
2024 Austrian MO Regional Competition, 1
Let $a$, $b$ and $c$ be real numbers larger than $1$. Prove the inequality $$\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.$$
When does equality hold?
[i](Karl Czakler)[/i]
2021 Junior Macedonian Mathematical Olympiad, Problem 4
Let $a$, $b$, $c$ be positive real numbers such that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} = \frac{27}{4}.$ Show that:
$$\frac{a^3+b^2}{a^2+b^2} + \frac{b^3+c^2}{b^2+c^2} + \frac{c^3+a^2}{c^2+a^2} \geq \frac{5}{2}.$$
[i]Authored by Nikola Velov[/i]
2023 Bulgaria National Olympiad, 6
In a class of $26$ students, everyone is being graded on five subjects with one of three possible marks. Prove that if $25$ of these students have received their marks, then we can grade the last one in such a way that their marks differ from these of any other student on at least two subjects.
2010 Math Prize For Girls Problems, 4
Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$. The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$? (The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.)
2025 India STEMS Category A, 6
Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots.
(Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$).
[i]Proposed by Malay Mahajan[/i]
2002 AIME Problems, 1
Given that
\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& \text{y is the number formed by reversing the digits of x; and}\\ &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of $z$ are possible?
2018 BMT Spring, 6
A triangle $T$ has all integer side lengths and at most one of its side lengths is greater than ten. What is the largest possible area of $T$ ?
2014 Stars Of Mathematics, 2
Determine all integers $n\geq 1$ for which the numbers $1,2,\ldots,n$ may be (re)ordered as $a_1,a_2,\ldots,a_n$ in such a way that the average $\dfrac {a_1+a_2+\cdots + a_k} {k}$ is an integer for all values $1\leq k\leq n$.
(Dan Schwarz)
2020 HMNT (HMMO), 8
A bar of chocolate is made of $10$ distinguishable triangles as shown below:
[center][img]https://cdn.artofproblemsolving.com/attachments/3/d/f55b0af0ce320fbfcfdbfab6a5c9c9306bfd16.png[/img][/center]
How many ways are there to divide the bar, along the edges of the triangles, into two or more contiguous pieces?
2012 National Olympiad First Round, 18
If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \text{None}$
1969 AMC 12/AHSME, 30
Let $P$ be a point of hypotenuse $AB$ (or its extension) of isosceles right triangle $ABC$. Let $s=AP^2+PB^2$. Then:
$\textbf{(A) }s<2CP^2\text{ for a finite number of positions of }P$
$\textbf{(B) }s<2CP^2\text{ for an infinite number of positions of }P$
$\textbf{(C) }s=2CP^2\text{ only if }P\text{ is the midpoint of }AB\text{ or an endpoint of }AB$
$\textbf{(D) }s=2CP^2\text{ always}$
$\textbf{(E) }s>2CP^2\text{ if }P\text{ is a trisection point of }AB$
2020 Latvia Baltic Way TST, 6
For a natural number $n \ge 3$ we denote by $M(n)$ the minimum number of unit squares that must be coloured in a $6 \times n$ rectangle so that any possible $2 \times 3$ rectangle (it can be rotated, but it must be contained inside and cannot be cut) contains at least one coloured unit square. Is it true that for every natural $n \ge 3$ the number $M(n)$ can be expressed as $M(n)=p_n+k_n^3$, where $p_n$ is a prime and $k_n$ is a natural number?
Kyiv City MO 1984-93 - geometry, 1989.8.5
The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.
1985 IMO Longlists, 60
The sequence $(s_n)$, where $s_n= \sum_{k=1}^n \sin k$ for $n = 1, 2,\dots$ is bounded. Find an upper and lower bound.
2012 Denmark MO - Mohr Contest, 1
Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$.
[img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]
2012 Centers of Excellency of Suceava, 1
Function ${{f\colon \mathbb[0, +\infty)}\to\mathbb[0, +\infty)}$ satisfies the condition $f(x)+f(y){\ge}2f(x+y)$ for all $x,y{\ge}0$.
Prove that $f(x)+f(y)+f(z){\ge}3f(x+y+z)$ for all $x,y,z{\ge}0$.
Mathematical induction?
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Azerbaijan Land of the Fire :lol:
2019 India National OIympiad, 6
Let $f$ be a function defined from $((x,y) : x,y$ real, $xy\ne 0)$ to the set of all positive real numbers such that
$ (i) f(xy,z)= f(x,z)\cdot f(y,z)$ for all $x,y \ne 0$
$ (ii) f(x,yz)= f(x,y)\cdot f(x,z)$ for all $x,y \ne 0$
$ (iii) f(x,1-x) = 1 $ for all $x \ne 0,1$
Prove that
$ (a) f(x,x) = f(x,-x) = 1$ for all $x \ne 0$
$(b) f(x,y)\cdot f(y,x) = 1 $ for all $x,y \ne 0$
The condition (ii) was left out in the paper leading to an incomplete problem during contest.