Found problems: 85335
2020 Moldova Team Selection Test, 9
Let $\Delta ABC$ be an acute triangle and $\Omega$ its circumscribed circle, with diameter $AP$. Points $E$ and $F$ are the orthogonal projections from $B$ on $AC$ and $AP$, points $M$ and $N$ are the midpoints of segments $EF$ and $CP$. Prove that $\angle BMN=90$.
1973 Putnam, A4
How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?
2008 District Olympiad, 3
Let $ A$ be a commutative unitary ring with an odd number of elements.
Prove that the number of solutions of the equation $ x^2 \equal{} x$ (in $ A$) divides the number of invertible elements of $ A$.
2023 CMIMC Algebra/NT, 9
Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$?
[i]Proposed by Giacomo Rizzo[/i]
2011 Indonesia TST, 3
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
2024 HMNT, 20
There exists a unique line tangent to the graph of $y=x^4-20x^3+24x^2-20x+25$ at two distinct points. Compute the product of the $x$-coordinates of the two tangency points.
2007 Baltic Way, 14
In a convex quadrilateral $ABCD$ we have $ADC = 90^{\circ}$. Let $E$ and $F$ be the projections of $B$ onto the lines $AD$ and $AC$, respectively. Assume that $F$ lies between $A$ and $C$, that $A$ lies between $D$ and $E$, and that the line $EF$ passes through the midpoint of the segment $BD$. Prove that the quadrilateral $ABCD$ is cyclic.
Kvant 2021, M2667
Does there exist a set $S$ of $100$ points in a plane such that the center of mass of any $10$ points in $S$ is also a point in $S$?
2005 Romania National Olympiad, 4
Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let
\[ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . \]
Prove that the following statements are equivalent:
a) $A$ is a field;
b) $M$ is not empty and the smallest element in $M$ is $2^n+1$.
[i]Marian Andronache[/i]
2007 ITest, 43
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers: \begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.
1992 IMTS, 1
The set $S$ consists of five integers. If pairs of distinct elements of $S$ are added, the following ten sums are obtained: 1967,1972,1973,1974,1975,1980,1983,1984,1989,1991. What are the elements of $S$?
2013 NIMO Summer Contest, 12
In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$.
[i]Proposed by Eugene Chen[/i]
2000 Stanford Mathematics Tournament, 3
A twelve foot tree casts a five foot shadow. How long is Henry's shadow (at the same time of day) if he is five and a half feet tall?
2012 China Team Selection Test, 3
$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.
2018 CCA Math Bonanza, TB4
Triangle $ABC$ is a triangle with side lengths $13$, $14$, and $15$. A point $Q$ is chosen uniformly at random in the interior of $\triangle{ABC}$. Choose a random ray (of uniformly random direction) with endpoint $Q$ and let it intersect the perimeter of $\triangle{ABC}$ at $P$. What is the expected value of $QP^2$?
[i]2018 CCA Math Bonanza Tiebreaker Round #4[/i]
2017 Ecuador Juniors, 4
Indicate whether it is possible to write the integers $1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of an regular octagon such that the sum of the numbers of any $3$ consecutive vertices is greater than:
a) $11$.
b) $13$.
2017 AIME Problems, 12
Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,...$ and a sequence of points on the circles $A_1,A_2,A_3,...$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r=\frac{11}{60}$, the distance from the center of $C_0$ to $B$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(6cm);
real r = 0.8;
pair nthCircCent(int n){
pair ans = (0, 0);
for(int i = 1; i <= n; ++i)
ans += rotate(90 * i - 90) * (r^(i - 1) - r^i, 0);
return ans;
}
void dNthCirc(int n){
draw(circle(nthCircCent(n), r^n));
}
dNthCirc(0);
dNthCirc(1);
dNthCirc(2);
dNthCirc(3);
dot("$A_0$", (1, 0), dir(0));
dot("$A_1$", nthCircCent(1) + (0, r), dir(135));
dot("$A_2$", nthCircCent(2) + (-r^2, 0), dir(0));
[/asy]
2023 Azerbaijan IMO TST, 1
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
2011 Indonesia TST, 3
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
1963 Putnam, A3
Find an integral formula for the solution of the differential equation
$$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$
for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$
LMT Team Rounds 2021+, B28
Maisy and Jeff are playing a game with a deck of cards with $4$ $0$’s, $4$ $1$’s, $4$ $2$’s, all the way up to $4$ $9$’s. You cannot tell apart cards of the same number. After shuffling the deck, Maisy and Jeff each take $4$ cards, make the largest $4$-digit integer they can, and then compare. The person with the larger $4$-digit integer wins. Jeff goes first and draws the cards $2,0,2,1$ from the deck. Find the number of hands Maisy can draw to beat that, if the order in which she draws the cards matters.
[i]Proposed by Richard Chen[/i]
2022 Bulgarian Autumn Math Competition, Problem 11.2
Given is a triangle $ABC$ and a circle through $A, B$. The perpendicular bisector of $AB$ meets the circle at $P, Q$, such that $AP>AQ$. Let $M$ be a point on the segment $AB$. The lines through $M$, parallel to $QA, QB$ meet $PB, PA$ at $R, S$. Prove that $MQ$ bisects $RS$.
2020 Silk Road, 1
Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.
2003 Czech And Slovak Olympiad III A, 3
A sequence $(x_n)_{n= 1}^{\infty}$ satisfies $x_1 = 1$ and for each $n > 1, x_n = \pm (n-1)x_{n-1} \pm (n-2)x_{n-2} \pm ... \pm 2x_2 \pm x_1$. Prove that the signs ” $\pm$” can be chosen so that $x_n \ne 12$ holds only for finitely many $n$.
2021 IMO Shortlist, A4
Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$