Found problems: 85335
1995 Polish MO Finals, 1
The positive reals $x_1, x_2, ... , x_n$ have harmonic mean $1$. Find the smallest possible value of $x_1 + \frac{x_2 ^2}{2} + \frac{x_3 ^3}{3} + ... + \frac{x_n ^n}{n}$.
2018 China National Olympiad, 1
Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying
$$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$
are both integers that are relatively prime to $p$. If $A_n$ is finite, let $f(n)$ denote $|A_n|$.
a) Prove that $A_n$ is finite if and only if $n \not = 2$.
b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that
$$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$
2014 ASDAN Math Tournament, 1
Alex gets $8$ points on an exam, while his friend gets $3$ times as many points as Alex. What is the average of their scores?
2018 Yasinsky Geometry Olympiad, 4
Let $I_a$ be the point of the center of an ex-circle of the triangle $ABC$, which touches the side $BC$ . Let $W$ be the intersection point of the bisector of the angle $\angle A$ of the triangle $ABC$ with the circumcircle of the triangle $ABC$. Perpendicular from the point $W$ on the straight line $AB$, intersects the circumcircle of $ABC$ at the point $P$. Prove, that if the points $B, P, I_a$ lie on the same line, then the triangle $ABC$ is isosceles.
(Mykola Moroz)
2013 BMT Spring, 4
Let $ABCD$ be a square with side length $2$, and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle?
2013 South East Mathematical Olympiad, 5
$f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]$. A integer $n$ is called [i]good[/i] if $f(x)=n$ has real root. How many good numbers are in $\{1,3,5,\dotsc,2013\}$?
2013 Online Math Open Problems, 3
A [i]palindromic table[/i] is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[ \begin{array}[h]{ccc}
O & M & O \\
N & M & N \\
O & M & O
\end{array} \]
How many palindromic tables are there that use only the letters $O$ and $M$? (The table may contain only a single letter.)
[i]Proposed by Evan Chen[/i]
2022 Yasinsky Geometry Olympiad, 3
Given a triangle $ABC$, in which the medians $BE$ and $CF$ are perpendicular. Let $M$ is the intersection point of the medians of this triangle, and $L$ is its Lemoine point (the intersection point of lines symmetrical to the medians with respect to the bisectors of the corresponding angles). Prove that $ML \perp BC$.
(Mykhailo Sydorenko)
1999 Spain Mathematical Olympiad, 1
The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.
2016 Kyiv Mathematical Festival, P1
Prove that for every positive integers $a$ and $b$ there exist positive integers $x$ and $y$ such that $\dfrac{x}{y+a}+\dfrac{y}{x+b}=\dfrac{3}{2}.$
2009 Mathcenter Contest, 3
Let $x,y,z>0$ Prove that $$\frac{x^2+2}{\sqrt{z^2+xy}}+\dfrac{y^2+2}{\sqrt{x ^2+yz}}+\dfrac{z^2+2}{\sqrt{y^2+zx}}\geq 6$$.
[i](nooonuii)[/i]
2004 Turkey Team Selection Test, 1
An $11\times 11$ chess board is covered with one $\boxed{ }$ shaped and forty $\boxed{ }\boxed{ }\boxed{ }$ shaped tiles. Determine the squares where $\boxed{}$ shaped tile can be placed.
2019 Flanders Math Olympiad, 1
Two touching balls with radii $a$ and $b$ are enclosed in a cylindrical tin of diameter $d$ . Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that $\sqrt{d} =\sqrt{a} +\sqrt{b}$
[img]https://1.bp.blogspot.com/-O4B3P3bghFs/Xy1fDv9zGkI/AAAAAAAAMSQ/ePLVnsXsRi0mz3SWBpIzfGdsizWoLmGVACLcBGAsYHQ/s0/flanders%2B2019%2Bp1.png[/img]
2012 Indonesia TST, 2
Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.
2022 Assam Mathematical Olympiad, 10
Let the vertices of the square $ABCD$ are on a circle of radius $r$ and with center $O$. Let $P, Q, R$ and $S$ are the mid points of $AB, BC, CD$ and $DA$ respectively. Then;
(a) Show that the quadrilateral $P QRS$ is a square.
(b) Find the distance from the mid point of $P Q$ to $O$.
2017-2018 SDPC, 4
Call a positive rational number in simplest terms [i]coddly[/i] if its numerator and denominator are both odd. Consider the equation $$2017= x_1\text{ }\square\text{ }x_2\text{ }\square\text{ }x_3\text{ }\ldots \text{ }\square \text{ }x_{2016} \text{ }\square \text{ }x_{2017},$$ where there are $2016$ boxes. We fill up the boxes randomly with the operations $+$, $-$, and $\times$. Compute the probability that there exists a solution in [b]distinct[/b] coddly numbers $(x_1,x_2, \ldots x_{2017})$ to the resulting equation.
1994 Putnam, 1
Find all positive integers that are within $250$ of exactly $15$ perfect squares.
2024 CMIMC Combinatorics and Computer Science, 3
Milo rolls five fair dice which have 4, 6, 8, 12, and 20 sides respectively (and each one is labeled $1$-$n$ for appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted.
[i]Proposed by Robert Trosten[/i]
1999 Croatia National Olympiad, Problem 1
Circles $k_1$ and $k_2$ with radii $r_1=6$ and $r_2=3$ are externally tangent and touch a circle $k$ with radius $r=9$ from inside. A common external tangent of $k_1$ and $k_2$ intersects $k$ at $P$ and $Q$. Determine the length of $PQ$.
1949-56 Chisinau City MO, 49
Prove the identity: $$\cos \frac{\pi}{7} \cdot \cos \frac{4\pi}{7} \cdot \cos \frac{5\pi}{7} = \frac{1}{8}$$
2014 Canada National Olympiad, 1
Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.
2021/2022 Tournament of Towns, P1
Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers?
[i]Boris Frenkin[/i]
1969 Putnam, A4
Show that
$$ \int_{0}^{1} x^{x} \, dx = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^n }.$$
1988 IMO Longlists, 11
Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$
2013 District Olympiad, 4
Consider the square $ABCD$ and the point $E$ inside the angle $CAB$, such that $\angle BAE =15^o$, and the lines $BE$ and $BD$ are perpendicular. Prove that $AE = BD$.