Found problems: 85335
2009 Today's Calculation Of Integral, 512
Evaluate $ \int_0^{n\pi} \sqrt{1\minus{}\sin t}\ dt\ (n\equal{}1,\ 2,\ \cdots).$
2017 Tuymaada Olympiad, 6
Let $\sigma(n) $ denote the sum of positive divisors of a number $n $. A positive integer $N=2^rb $ is given,where $r $ and $b $ are positive integers and $b $ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and $\sigma (b) $ are coprime.
Tuymaada Q6 Juniors
1992 IMO Longlists, 46
Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$
2018 AMC 12/AHSME, 9
What is \[ \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? \]
$
\textbf{(A) }100,100 \qquad
\textbf{(B) }500,500\qquad
\textbf{(C) }505,000 \qquad
\textbf{(D) }1,001,000 \qquad
\textbf{(E) }1,010,000 \qquad
$
2021 MMATHS, 1
Let $a,b,c$ be the roots of the polynomial $x^3 - 20x^2 + 22.$ Find \[\frac{bc}{a^2}+\frac{ac}{b^2}+\frac{ab}{c^2}.\]
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2009 IMO Shortlist, 4
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
2004 Harvard-MIT Mathematics Tournament, 8
Let $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$.
VMEO III 2006 Shortlist, G1
Given a circle $(O)$ and a point $P$ outside that circle. $M$ is a point running on the circle $(O)$. The circle with center $I$ and diameter $PM$ intersects circle $(O)$ again at $N$. The tangent of $(I)$ at $P$ intersects $MN$ at $Q$. The line through $Q$ perpendicular to $PO$ intersects $PM$ at $ A$. $AN$ intersects $(O)$ further at $ B$. $BM$ intersects $PO$ at $C$. Prove that $AC$ is perpendicular to $OQ$.
2008 ISI B.Stat Entrance Exam, 5
Suppose $ABC$ is a triangle with inradius $r$. The incircle touches the sides $BC, CA,$ and $AB$ at $D,E$ and $F$ respectively. If $BD=x, CE=y$ and $AF=z$, then show that
\[r^2=\frac{xyz}{x+y+z}\]
2021 Junior Balkаn Mathematical Olympiad, 3
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$.
Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.
1992 Tournament Of Towns, (345) 3
Do there exist two polynomials $P(x)$ and $Q(x)$ with integer coefficients such that
$$(P-Q)(x), \,\,\,\, P(x) \,\,\,\, and \,\,\,\,(P+Q)(x)$$
are squares of polynomials (and $Q$ is not equal to $cP$, where $c$ is a real number)?
(V Prasolov)
2022 DIME, 4
Given a regular hexagon $ABCDEF$, let point $P$ be the intersection of lines $BC$ and $DE$, and let point $Q$ be the intersection of lines $AP$ and $CD$. If the area of $\triangle QEP$ is equal to $72$, find the area of regular hexagon $ABCDEF$.
[i]Proposed by [b]DeToasty3[/b][/i]
1998 Iran MO (3rd Round), 2
Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that
\[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC.
\]
Prove that
\[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1.
\]
2001 Moldova National Olympiad, Problem 3
A line $d_i~(i=1,2,3)$ intersects two opposite sides of a square $ABCD$ at points $M_i$ and $N_i$. Prove that if $M_1N_1=M_2N_2=M_3N_3$, then two of the lines $d_i$ are either parallel or perpendicular.
Denmark (Mohr) - geometry, 2000.1
The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$.
[img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]
2018 China Girls Math Olympiad, 5
Let $\omega \in \mathbb{C}$, and $\left | \omega \right | = 1$. Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$.
2010 Mathcenter Contest, 1
A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition:
\[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\]
Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds:
\[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]
2005 Baltic Way, 8
Consider a $25 \times 25$ grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?
2022 Portugal MO, 5
In a badminton competition, $16$ players participate, of which $10$ are professionals and $6$ are amateurs. In the first phase, eight games are drawn. Among the eight winners of these games, four games are drawn. The four winners qualify for the semi-finals of the competition. Assuming that, whenever a professional player and an amateur play each other, the professional wins the game, what is the probability that an amateur player will reach the semi-finals of the competition?
2009 ISI B.Math Entrance Exam, 1
Let $x,y,z$ be non-zero real numbers. Suppose $\alpha, \beta, \gamma$ are complex numbers such that $|\alpha|=|\beta|=|\gamma|=1$. If $x+y+z=0=\alpha x+\beta y+\gamma z$, then prove that $\alpha =\beta =\gamma$.
1898 Eotvos Mathematical Competition, 1
Determine all positive integers $n$ for which $2^n + 1$ is divisible by $3$.
2018 Switzerland - Final Round, 2
Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take:
$$\frac{a}{gcd\,\,(a + b, a - c)}
+
\frac{b}{gcd\,\,(b + c, b - a)}
+
\frac{c}{gcd\,\,(c + a, c - b)}.$$
.
Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.
2023 CMIMC Team, 10
Consider the set of all permutations, $\mathcal{P}$, of $\{1,2,\ldots,2022\}$. For permutation $P\in \mathcal{P}$, let $P_1$ denote the first element in $P$. Let $\text{sgn}(P)$ denote the sign of the permutation. Compute the following number modulo 1000: $$\displaystyle\sum_{P\in\mathcal{P}}\dfrac{P_1\cdot\text{sgn}(P)^{P_1}}{2020!}.$$
(The [i]sign[/i] of a permutation $P$ is $(-1)^k$, where $k$ is the minimum number of two-element swaps needed to reach that permutation).
[i]Proposed by Nairit Sarkar[/i]
2006 MOP Homework, 4
Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.
2010 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle in which $\angle A = 60^\circ$. Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$. Let $M$ be the reflection of $A$ in line $EF$. Prove that $M$ lies on $BC$.