Found problems: 85335
1999 China National Olympiad, 1
Let $m$ be a positive integer. Prove that there are integers $a, b, k$, such that both $a$ and $b$ are odd, $k\geq0$ and
\[2m=a^{19}+b^{99}+k\cdot2^{1999}\]
1989 Putnam, B6
Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum
$$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.
PEN J Problems, 7
Show that if the equation $\phi(x)=n$ has one solution, it always has a second solution, $n$ being given and $x$ being the unknown.
2015 Purple Comet Problems, 5
The diagram below shows a rectangle with one side divided into seven equal segments and the opposite
side divided in half. The rectangle has area 350. Find the area of the shaded region. For Diagram go to purplecomet.org/welcome/practice, the $2015$ middle school contest, and #5.
2021 MMATHS, 6
Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$, then find $a + b$.
[i]Proposed by Vismay Sharan[/i]
2023 Math Prize for Girls Problems, 7
An arithmetic expression is created by inserting either a plus sign or a multiplication sign in each of the 11 spaces between consecutive $\sqrt{3}$’s in a row of twelve $\sqrt{3}$’s. The signs are chosen uniformly and independently at random. What is the probability that the resulting expression evaluates to $12\sqrt{3}$?
2023 India IMO Training Camp, 1
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations, for any $x,y \in \mathbb{N}$.
2017 Baltic Way, 13
Let $ABC$ be a triangle in which $\angle ABC = 60^{\circ}$. Let $I$ and $O$ be the incentre and circumcentre of $ABC$, respectively. Let $M$ be the midpoint of the arc $BC$ of the circumcircle of $ABC$, which does not contain the point $A$. Determine $\angle BAC$ given that $MB = OI$.
2022 HMNT, 7
In circle $\omega$, two perpendicular chords intersect at a point $P$. The two chords have midpoints $M_1$ and $M_2$ respectively, such that $PM_1=15$ and $PM_2=20$. Line $M_1M_2$ intersects $\omega$ at points $A$ and $B$, with $M_1$ between $A$ and $M_2$. Compute the largest possible value of $BM_2-AM_1$.
2016 CMIMC, 3
Let $\varepsilon$ denote the empty string. Given a pair of strings $(A,B)\in\{0,1,2\}^*\times\{0,1\}^*$, we are allowed the following operations:
\[\begin{cases}
(A,1)\to(A0,\varepsilon)\\
(A,10)\to(A00,\varepsilon)\\
(A,0B)\to(A0,B)\\
(A,11B)\to(A01,B)\\
(A,100B)\to(A0012,1B)\\
(A,101B)\to(A00122,10B)
\end{cases}\]
We perform these operations on $(A,B)$ until we can no longer perform any of them. We then iteratively delete any instance of $20$ in $A$ and replace any instance of $21$ with $1$ until there are no such substrings remaining. Among all binary strings $X$ of size $9$, how many different possible outcomes are there for this process performed on $(\varepsilon,X)$?
MathLinks Contest 5th, 3.1
Let $\{x_n\}_n$ be a sequence of positive rational numbers, such that $x_1$ is a positive integer, and for all positive integers $n$.
$x_n = \frac{2(n - 1)}{n} x_{n-1}$, if $x_{n_1} \le 1$
$x_n = \frac{(n - 1)x_{n-1} - 1}{n}$ , if $x_{n_1} > 1$.
Prove that there exists a constant subsequence of $\{x_n\}_n$.
2015 Brazil Team Selection Test, 4
Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
[i]Proposed by Jack Edward Smith, UK[/i]
2019 Tournament Of Towns, 6
A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit cube.
a) Prove that there exists a regular subset consisting of $2N^2$ needles such that all of them have either the same direction or two different directions.
b) What is the maximum size of a regular subset that does exist for sure?
(Nikita Gladkov, Alexandr Zimin)
2020 Caucasus Mathematical Olympiad, 3
Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.
2023 Korea Junior Math Olympiad, 7
Find the smallest positive integer $N$ such that there are no different sets $A, B$ that satisfy the following conditions. (Here, $N$ is not a power of $2$. That is, $N \neq 1, 2^1, 2^2, \dots$.)
[list]
[*] $A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}$
[*] $|A| = |B| \geq 1$
[*] Sum of elements in $A$ and sum of elements in $B$ are equal.
[/list]
1994 All-Russian Olympiad Regional Round, 10.2
The equation $ x^2 \plus{} ax \plus{} b \equal{} 0$ has two distinct real roots. Prove that the equation $ x^4 \plus{} ax^3 \plus{} (b \minus{} 2)x^2 \minus{} ax \plus{} 1 \equal{} 0$ has four distinct real roots.
1997 Estonia National Olympiad, 2
Side lengths $a,b,c$ of a triangle satisfy $\frac{a^3+b^3+c^3}{a+b+c}= c^2$. Find the measure of the angle opposite to side $c$.
2017 Purple Comet Problems, 17
Let $a_0$, $a_1$, ..., $a_6$ be real numbers such that $a_0 + a_1 + ... + a_6 = 1$ and
$$a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}$$
$$a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}$$
$$a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}$$
$$a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}$$
The value of $a_0$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2022 Malaysia IMONST 2, 6
A football league has $n$ teams. Each team plays one game with every other team. Each win is awarded $2$ points, each tie $1$ point, and each loss $0$ points.
After the league is over, the following statement is true: for every subset $S$ of teams in the league, there is a team (which may or may not be in $S$) such that the total points the team obtained by playing all the teams in $S$ is odd.
Prove that $n$ is even.
2016 AIME Problems, 4
A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60^\circ$. Find $h^2$.
1998 All-Russian Olympiad Regional Round, 8.2
Given a parallelogram ABCD, let M and N be the midpoints of the sides BC and CD.
Can the lines AM, AN divide the angle BAD into three equal angles?
2011 JHMT, 7
Let $ABCD$ be a cyclic quadrilateral with $AB = 6$, $BC = 12$, $CD = 3$, and $DA = 6$. Let $E, F$ be the intersection of lines $AB$ and $CD$, lines $AD$ and $BC$, respectively. Find $EF$.
2003 Indonesia MO, 3
Find all real numbers $x$ such that $\left\lfloor x^2 \right\rfloor + \left\lceil x^2 \right\rceil = 2003$.
2016 Kosovo Team Selection Test, 3
If quadratic equations $x^2+ax+b=0$ and $x^2+px+q=0$ share one similar root then find quadratic equation for which has roots of other roots of both quadratic equations .
2021 Saudi Arabia JBMO TST, 2
In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.