Found problems: 85335
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?
2005 Dutch Mathematical Olympiad, 1
In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5?
2015 Princeton University Math Competition, A3
Let $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $ABC$ at $D \neq C $. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $AIC$ at $Y$ outside $\omega_1$. She notices that she can construct a right triangle with side lengths $ID$, $DX$, and $XY$. Determine, with proof, the length of $IY$.
2022 Math Prize for Girls Olympiad, 2
Determine, with proof, whether or not there exists a [i]non-isosceles[/i] trapezoid $ABCD$ such that the lengths $AC$ and $BD$ both lie in the set $\{ DA+AB, AB+BC, BC+CD, CD+DA, AB+CD, BC+DA \}$.
1997 National High School Mathematics League, 12
Let $a=\lg z+\lg\left[x(yz)^{-1}+1\right],b=\lg x^{-1}+\lg(xyz+1),c=\lg y+\lg\left[(xyz)^{-1}+1\right]$, if $M=\max\{a,b,c\}$, then the minumum value of $M$ is________.
2010 Purple Comet Problems, 6
Evaluate the sum $1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 \cdots + 208 + 209 - 210.$
1992 IMO Shortlist, 15
Does there exist a set $ M$ with the following properties?
[i](i)[/i] The set $ M$ consists of 1992 natural numbers.
[i](ii)[/i] Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$
1996 Czech and Slovak Match, 3
The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a$\sqrt{17}$. Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.
2009 National Olympiad First Round, 31
For all $ |x| \ge n$, the inequality $ |x^3 \plus{} 3x^2 \minus{} 33x \minus{} 3| \ge 2x^2$ holds. Integer $ n$ can be at least ?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 5$