Found problems: 85335
2014 Harvard-MIT Mathematics Tournament, 31
Compute \[\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.\]
2021 Cyprus JBMO TST, 4
Let $\triangle AB\varGamma$ be an acute-angled triangle with $AB < A\varGamma$, and let $O$ be the center of the circumcircle of the triangle. On the sides $AB$ and $A \varGamma$ we select points $T$ and $P$ respectively such that $OT=OP$. Let $M,K$ and $\varLambda$ be the midpoints of $PT,PB$ and $\varGamma T$ respectively. Prove that $\angle TMK = \angle M\varLambda K$.
2019-IMOC, G2
Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic.
[img]https://3.bp.blogspot.com/-v7D_A66nlD0/XnYNJussW9I/AAAAAAAALeQ/q6DMQ7w6QtI5vLwBcKqp4010c3XTCj3BgCK4BGAYYCw/s1600/imoc2019g2.png[/img]
2011 All-Russian Olympiad, 4
Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle.
[i]M. Kungojin[/i]
2007 Tournament Of Towns, 3
Two players in turns color the squares of a $4 \times 4$ grid, one square at the time. Player loses if after his move a square of $2\times2$ is colored completely. Which of the players has the winning strategy, First or Second?
[i](3 points)[/i]
2012 Iran Team Selection Test, 1
Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder.
[i]Proposed by Yahya Motevassel[/i]
2016 ASDAN Math Tournament, 3
Julia adds up the numbers from $1$ to $2016$ in a calculator. However, every time she inputs a $2$, the calculator malfunctions and inputs a $3$ instead (for example, when Julia inputs $202$, the calculator inputs $303$ instead). How much larger is the total sum returned by the broken calculator? (No $2$s are replaced by $3$s in the output, and the calculator only malfunctions while Julia is inputting numbers.)
1978 IMO Shortlist, 3
Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.
1995 May Olympiad, 2
The owner of a hardware store bought a quantity of screws in closed boxes and sells the screws separately. He never has more than one open box. At the end of the day Monday there are $2208$ screws left, at the end of the day Tuesday there are still $1616$ screws and at the end of Wednesday there are still $973$ screws. To control the employees, every night he writes down the number of screws that are in the only open box. The amount entered on Tuesday is double that of Monday. How many screws are there in each closed box if it is known that they are less than $500$?
2007 Today's Calculation Of Integral, 234
For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
Evaluate $ \int_0^{100\pi } f(x)\ dx.$
2016 Abels Math Contest (Norwegian MO) Final, 1
A "[size=100][i]walking sequence[/i][/size]" is a sequence of integers with $a_{i+1} = a_i \pm 1$ for every $i$ .Show that there exists a sequence $b_1, b_2, . . . , b_{2016}$ such that for every walking sequence $a_1, a_2, . . . , a_{2016}$ where $1 \leq a_i \leq1010$, there is for some $j$ for which $a_j = b_j$ .
BIMO 2021, 2
Let $ABC$ be a triangle with incircle centered at $I$, tangent to sides $AC$ and $AB$ at $E$ and $F$ respectively. Let $N$ be the midpoint of major arc $BAC$. Let $IN$ intersect $EF$ at $K$, and $M$ be the midpoint of $BC$. Prove that $KM\perp EF$.
Kvant 2020, M2607
Let $n$ be a natural number. The set $A{}$ of natural numbers has the following property: for any natural number $m\leqslant n$ in the set $A{}$ there is a number divisible by $m{}$. What is the smallest value that the sum of all the elements of the set $A{}$ can take?
[i]Proposed by A. Kuznetsov[/i]
2022 Czech and Slovak Olympiad III A, 2
We say that a positive integer $k$ is [i]fair [/i] if the number of $2021$-digit palindromes that are a multiple of $k$ is the same as the number of $2022$-digit palindromes that are a multiple of $k$. Does the set $M = \{1, 2,..,35\}$ contain more numbers that are fair or those that are not fair?
(A palindrome is an integer that reads the same forward and backward.)
[i](David Hruska)[/i]
2025 Malaysian IMO Training Camp, 3
Given a triangle $ABC$ with $M$ the midpoint of minor arc $BC$. Let $H$ be the feet of altitude from $A$ to $BC$. Let $S$ and $T$ be the reflections of $B$ and $C$ with respect to line $AM$. Suppose the circle $(HST)$ meets $BC$ again at a point $P$. Prove that $\angle AMP = 90^\circ$.
[i](Proposed by Tan Rui Xuen)[/i]
2008 AMC 10, 8
Heather compares the price of a new computer at two different stores. Store A offers $ 15\%$ off the sticker price followed by a $ \$90$ rebate, and store B offers $ 25\%$ off the same sticker price with no rebate. Heather saves $ \$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?
$ \textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$
2015 ASDAN Math Tournament, 29
Suppose that the following equations hold for positive integers $x$, $y$, and $n$, where $n>18$:
\begin{align*}
x+3y&\equiv7\pmod{n}\\
2x+2y&\equiv18\pmod{n}\\
3x+y&\equiv7\pmod{n}
\end{align*}
Compute the smallest nonnegative integer $a$ such that $2x\equiv a\pmod{n}$.
2020 Italy National Olympiad, #1
Let $\omega$ be a circle and let $A,B,C,D,E$ be five points on $\omega$ in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$.
a) Prove that the lines $AC$ and $DE$ are parallel
b) Prove that $AE=CD$
2014 Iran MO (3rd Round), 2
We say two sequence of natural numbers A=($a_1,...,a_n$) , B=($b_1,...,b_n$)are the exchange and we write $A\sim B$.
if $503\vert a_i - b_i$ for all $1\leq i\leq n$.
also for natural number $r$ : $A^r$ = ($a_1^r,a_2^r,...,a_n^r$).
Prove that there are natural number $k,m$ such that :
$i$)$250 \leq k $
$ii$)There are different permutations $\pi _1,...,\pi_k$ from {$1,2,3,...,502$} such that for $1\leq i \leq k-1$ we have $\pi _i^m\sim \pi _{i+1}$
(15 points)
2014 Junior Regional Olympiad - FBH, 5
How many are there $4$ digit numbers such that they have two odd digits and two even digits
Estonia Open Senior - geometry, 2005.2.4
Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} > \sin \frac{\gamma}{2}$.
1996 Iran MO (3rd Round), 2
Let $ABCD$ be a convex quadrilateral. Construct the points $P,Q,R,$ and $S$ on continue of $AB,BC,CD,$ and $DA$, respectively, such that
\[BP=CQ=DR=AS.\]
Show that if $PQRS$ is a square, then $ABCD$ is also a square.
2016 Romania Team Selection Tests, 2
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
2001 Baltic Way, 17
Let $n$ be a positive integer. Prove that at least $2^{n-1}+n$ numbers can be chosen from the set $\{1, 2, 3,\ldots ,2^n\}$ such that for any two different chosen numbers $x$ and $y$, $x+y$ is not a divisor of $x\cdot y$.
2021 Math Prize for Girls Problems, 12
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?