Found problems: 85335
2014 Contests, 3
Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.
2005 Federal Math Competition of S&M, Problem 2
Let $ABC$ be an acute triangle. Circle $k$ with diameter $AB$ intersects $AC$ and $BC$ again at $M$ and $N$ respectively. The tangents to $k$ at $M$ and $N$ meet at point $P$. Given that $CP=MN$, determine $\angle ACB$.
1963 AMC 12/AHSME, 9
In the expansion of $\left(a-\dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\dfrac{1}{2}}$ is:
$\textbf{(A)}\ -7 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ -21 \qquad
\textbf{(D)}\ 21 \qquad
\textbf{(E)}\ 35$
2005 Silk Road, 2
Find all $(m,n) \in \mathbb{Z}^2$ that we can color each unit square of $m \times n$ with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.
2012 HMNT, 5
Let $\pi$ be a randomly chosen permutation of the numbers from $1$ through $2012$. Find the probability that$$ \pi (\pi(2012)) = 2012.$$
2021 Junior Balkan Team Selection Tests - Romania, P4
Let $n\geq 2$ be a positive integer. On an $n\times n$ board, $n$ rooks are placed in such a manner that no two attack each other. All rooks move at the same time and are only allowed to move in a square adjacent to the one in which they are located. Determine all the values of $n$ for which there is a placement of the rooks so that, after a move, the rooks still do not attack each other.
[i]Note: Two squares are adjacent if they share a common side.[/i]
2023 Iran MO (3rd Round), 3
In triangle $\triangle ABC$ points $M,N$ lie on $BC$ st : $\angle BAM= \angle MAN= \angle NAC$ . Points $P,Q$ are on the angle bisector of $BAC$, on the same side of $BC$ as A , st :
$$\frac{1}{3} \angle BAC = \frac{1}{2} \angle BPC = \angle BQC$$
Let $E = AM \cap CQ$ and $F = AN \cap BQ$ . Prove that the common tangents to $(EPF), (EQF)$ and the circumcircle of $\triangle ABC$ , are concurrent.
2013 Kosovo National Mathematical Olympiad, 4
Let be $n$ positive integer than calculate:
$1\cdot 1!+2\cdot2!+...+n\cdot n!$
2004 Bulgaria Team Selection Test, 1
The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.
1995 IberoAmerican, 2
The incircle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. Let the line $AD$ intersect this incircle of triangle $ABC$ at a point $X$ (apart from $D$). Assume that this point $X$ is the midpoint of the segment $AD$, this means, $AX = XD$. Let the line $BX$ meet the incircle of triangle $ABC$ at a point $Y$ (apart from $X$), and let the line $CX$ meet the incircle of triangle $ABC$ at a point $Z$ (apart from $X$). Show that $EY = FZ$.
2022-IMOC, G4
Let $\vartriangle ABC$ be an acute triangle with circumcircle $\Omega$. A line passing through $A$ perpendicular to $BC$ meets $\Omega$ again at $D$. Draw two circles $\omega_b$, $\omega_c$ with $B, C$ as centers and $BD$, $CD$ as radii, respectively, and they intersect $AB$, $AC$ at $E, F,$ respectively. Let $K\ne A$ be the second intersection of $(AEF)$ and $\Omega$, and let $\omega_b$, $\omega_c$ intersect $KB$, $KC$ at $P, Q$, respectively. The circumcenter of triangle $DP Q$ is $O$, prove that $K, O, D$ are collinear.
[i]proposed by Li4[/i]
2010 Purple Comet Problems, 11
There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats so that neither little boy sits to the side of either little girl. In how many different ways can these six people be seated?
2019 Ramnicean Hope, 1
Solve in the reals the equation $ \sqrt[3]{x^2-3x+4} +\sqrt[3]{-2x+2} +\sqrt[3]{-x^2+5x+2} =2. $
[i]Ovidiu Țâțan[/i]
2023 MOAA, 6
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$.
[i]Proposed by Harry Kim[/i]
2002 Czech and Slovak Olympiad III A, 4
Find all pairs of real numbers $a, b$ for which the equation in the domain of the real numbers
\[\frac{ax^2-24x+b}{x^2-1}=x\]
has two solutions and the sum of them equals $12$.
2016 IMO Shortlist, G2
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
[i]Proposed by Evan Chen, Taiwan[/i]
1991 Bundeswettbewerb Mathematik, 4
Given wo non-negative integers $a$ and $b$, one of them is odd and the other one even. By the following rule we define two sequences $(a_n),(b_n)$:
\[ a_0 = a, \quad a_1 = b, \quad a_{n+1} = 2a_n - a_{n-1} + 2 \quad (n = 1,2,3, \ldots)\]
\[ b_0 = b, \quad b_1 = a, \quad b_{n+1} = 2a_n - b_{n-1} + 2 \quad (n = 1,2,3, \ldots)\]
Prove that none of these two sequences contain a negative element if and only if we have $|\sqrt{a} - \sqrt{b}| \leq 1$.
1969 IMO Shortlist, 31
$(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$
Putnam 1939, A5
Do either $(1)$ or $(2)$
$(1)$ $x$ and $y$ are functions of $t.$ Solve $x' = x + y - 3, y' = -2x + 3y + 1,$ given that $x(0) = y(0) = 0.$
$(2)$ A weightless rod is hinged at $O$ so that it can rotate without friction in a vertical plane. A mass $m$ is attached to the end of the rod $A,$ which is balanced vertically above $O.$ At time $t = 0,$ the rod moves away from the vertical with negligible initial angular velocity. Prove that the mass first reaches the position under $O$ at $t = \sqrt{(\frac{OA}{g})} \ln{(1 + sqrt(2))}.$
MMPC Part II 1996 - 2019, 2016.3
This problem is about pairs of consecutive whole numbers satisfying the property that one of the numbers is a perfect square and the other one is the double of a perfect square.
(a) The smallest such pairs are $(0,1)$ and $(8,9)$, Indeed $0=2 \cdot 0^2$ and $1=1^2$; $8=2 \cdot 2^2$ and $9=3^2$. Show that there are infinitely many pairs of the form $(2a^2,b^2)$ where the smaller number is the double of a perfect square satisfying the given property.
(b) Find a pair of integers satisfying the property that is not in the form given in the first part, that is, find a
pair of integers such that the smaller one is a perfect square and the larger one is the double of a perfect
square.
1958 November Putnam, A5
Show that the number of non-zero integers in the expansion of the $n$-th order determinant having zeroes in the main diagonal and ones elsewhere is
$$n ! \left(1- \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^{n}}{n!} \right) .$$
2019 Philippine MO, 2
Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$.
2005 MOP Homework, 1
Let $a0$, $a1$, ..., $a_n$ be integers, not all zero, and all at least $-1$. Given that $a_0+2a_1+2^2a_2+...+2^na_n =0$, prove that $a_0+a_1+...+a_n>0$.
1988 Irish Math Olympiad, 10
Let $0\le x\le 1$. Show that if $n$ is any positive integer, then $$(1+x)^n\ge (1-x)^n+2nx(1-x^2)^{\frac{n-1}{2}}$$.
1999 Hungary-Israel Binational, 2
$ 2n\plus{}1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?