Found problems: 85335
1989 IMO Longlists, 30
Let $ ABC$ be an equilateral triangle. Let $ D,E, F,M,N,$ and $ P$ be the mid-points of $ BC, CA, AB, FD, FB,$ and $ DC$ respectively.
[b](a)[/b] Show that the line segments $ AM,EN,$ and $ FP$ are concurrent.
[b](b)[/b] Let $ O$ be the point of intersection of $ AM,EN,$ and $ FP.$ Find $ OM : OF : ON : OE : OP : OA.$
2002 German National Olympiad, 3
Prove that for all primes $p$ true is equality
$$\sum_{k=1}^{p-1}\left\lfloor\frac{k^3}{p}\right\rfloor=\frac{(p-2)(p-1)(p+1)}{4}$$
2010 IberoAmerican, 2
Determine if there are positive integers $a, b$ such that all terms of the sequence defined by
\[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.
V Soros Olympiad 1998 - 99 (Russia), 9.2
Solve the equation $x^4 + 4x^3 - 8x + 4 = 0$.
2004 Thailand Mathematical Olympiad, 5
Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =| 2nx - n - 2n^2|$
2002 Tournament Of Towns, 6
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
1991 Arnold's Trivium, 88
How many figures can be obtained by intersecting the infinite-dimensional cube $|x_k| \le 1$, $k = 1,2,\ldots$ with a two-dimensional plane?
2016 AIME Problems, 11
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.
2024 Bosnia and Herzegovina Junior BMO TST, 2.
Determine all $x$, $y$, $k$ and $n$ positive integers such that:
$10^x$ + $10^y$ + $n!$ = $2024^k$
2006 Tournament of Towns, 3
On sides $AB$ and $BC$ of an acute triangle $ABC$ two congruent rectangles $ABMN$ and $LBCK$ are constructed (outside of the triangle), so that $AB = LB$. Prove that straight lines $AL, CM$ and $NK$ intersect at the same point.
[i](5 points)[/i]
2004 Mediterranean Mathematics Olympiad, 4
Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and
\[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\]
If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.
1991 Spain Mathematical Olympiad, 1
In the coordinate plane, consider the set of all segments of integer lengths whose endpoints have integer coordinates. Prove that no two of these segments form an angle of $45^o$. Are there such segments in coordinate space?
1940 Eotvos Mathematical Competition, 2
Let $m$ and $n$ be distinct positive integers. Prove that $2^{2^m} + 1$ and $2^{2^n} + 1$ have no common divisor greater than $1$.
2005 National High School Mathematics League, 8
$f(x)$ is a decreasing function defined on $(0,+\infty)$, if $f(2a^2+a+1)<f(3a^2-4a+1)$, then the range value of $a$ is________.
2015 HMNT, 2
Bassanio has three red coins, four yellow coins, and five blue coins. At any point, he may give Shylock any two coins of different colors in exchange for one coin of the other color; for example, he may give Shylock one red coin and one blue coin, and receive one yellow coin in return. Bassanio wishes to end with coins that are all the same color, and he wishes to do this while having as many coins as possible. How many coins will he end up with, and what color will they be?
1987 Putnam, B4
Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist.
2021 AIME Problems, 11
Let $ABCD$ be a cyclic quadrilateral with $AB=4,BC=5,CD=6,$ and $DA=7$. Let $A_1$ and $C_1$ be the feet of the perpendiculars from $A$ and $C$, respectively, to line $BD,$ and let $B_1$ and $D_1$ be the feet of the perpendiculars from $B$ and $D,$ respectively, to line $AC$. The perimeter of $A_1B_1C_1D_1$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2024/2025 TOURNAMENT OF TOWNS, P1
Baron Munchausen took several cards and wrote a positive integer on each one (some numbers may be the same). The baron reports that he has used only two distinct digits to do that. He also reports that among the leftmost digits of the sums of integers on each pair of these cards there are all the digits from 1 to 9 . Can it occur that the baron is right? Maxim Didin
2014 Portugal MO, 1
The ship [i]Meridiano do Bacalhau[/i] does its fishing business during $64$ days. Each day the capitain chooses a direction which may be either north or south and the ship sails that direction in that day. On the first day of business the ship sails $1$ mile, on the second day sails $2$ miles; generally, on the $n$-th day it sails $n$ miles. After of the $64$-th day, the ship was $2014$ miles north from its initial position. What is the greatest number of days that the ship could have sailed south?
2024 All-Russian Olympiad Regional Round, 9.9
An equilateral triangle $T$ with side $111$ is partitioned into small equilateral triangles with side $1$ using lines parallel to the sides of $T$. Every obtained point except the center of $T$ is marked. A set of marked points is called $\textit{linear}$ if the points lie on a line, parallel to a side of $T$ (among the drawn ones). In how many ways we can split the marked point into $111$ $\textit{linear}$ sets?
2017 China Northern MO, 6
Define $S_r(n)$: digit sum of $n$ in base $r$. For example, $38=(1102)_3,S_3(38)=1+1+0+2=4$. Prove:
[b](a)[/b] For any $r>2$, there exists prime $p$, for any positive intenger $n$, $S_{r}(n)\equiv n\mod p$.
[b](b)[/b] For any $r>1$ and prime $p$, there exists infinitely many $n$, $S_{r}(n)\equiv n\mod p$.
2010 Romania Team Selection Test, 1
Each point of the plane is coloured in one of two colours. Given an odd integer number $n \geq 3$, prove that there exist (at least) two similar triangles whose similitude ratio is $n$, each of which has a monochromatic vertex-set.
[i]Vasile Pop[/i]
2007 Princeton University Math Competition, 10
Find the real root of $x^5+5x^3+5x-1$. Hint: Let $x = u+k/u$.
1968 IMO, 4
Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.
2013 Online Math Open Problems, 21
Dirock has a very neat rectangular backyard that can be represented as a $32\times 32$ grid of unit squares. The rows and columns are each numbered $1,2,\ldots, 32$. Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by $3$. Dirock would like to build a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of the yard such that
[list] [*] The fence does not pass through any grid squares containing rocks; [*] The interior of the fence contains exactly 5 rocks. [/list]
In how many ways can this be done?
[i]Ray Li[/i]