Found problems: 85335
In triangle $ABC, \angle B = 2\angle C$. Points $P$ and $Q$ on the medial perpendicular to $CB$ are such that $\angle CAP = \angle PAQ = \angle QAB = \frac{\angle A}{3}$ . Prove that $Q$ is the circumcenter of triangle $CPB$.
Consider $4n$ points in the plane such that no three of them are collinear ($n\geq 1$). Show that the set of centroids of all the triangles formed by any three of these points contains at least $4n$ elements.
[i]Radu Bumbăcea[/i]
Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds:
\[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]
Let $ABC$ be the right-angled isosceles triangle whose equal sides have length 1. $P$ is a point on the hypotenuse, and the feet of the perpendiculars from $P$ to the other sides are $Q$ and $R$. Consider the areas of the triangles $APQ$ and $PBR$, and the area of the rectangle $QCRP$. Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2/9$.
Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$ and $b^2+4a$ are both perfect squares.
A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides. What is the greatest possible area of the overlap of folded and unfolded parts of the triangle?
Cyclic quadrilateral $ABCD$ is such that $\angle BAD = 2\angle ADC$ and $CD = 2BC$. Let $H$ be the projection of $C$ onto $AD$. Prove that $BH \parallel CD$.
[i]Proposed by Fedir Yudin, Anton Trygub[/i]
Every side of a right triangle is an integer when measured in cm, and the difference between the hypotenuse and one of the legs is $75$ cm. What is the smallest possible value of its perimeter?
For $0 \leq p \leq 1/2,$ let $X_1, X_2, \ldots$ be independent random variables such that
$$X_i=\begin{cases}
1 & \text{with probability } p, \\
-1 & \text{with probability } p, \\
0 & \text{with probability } 1-2p,
\end{cases}
$$
for all $i \geq 1.$ Given a positive integer $n$ and integers $b,a_1, \ldots, a_n,$ let $P(b, a_1, \ldots, a_n)$ denote the probability that $a_1X_1+ \ldots + a_nX_n=b.$ For which values of $p$ is it the case that $$P(0, a_1, \ldots, a_n) \geq P(b, a_1, \ldots, a_n)$$ for all positive integers $n$ and all integers $b, a_1,\ldots, a_n?$
Find, with argument, the integer solutions of the equation
\[3z^2 = 2x^3 + 385x^2 + 256x - 58195.\]
Find all solutions to $3^x-9^{x-1} = 2.$
Medians $BK{}$ and $CN{}$ of triangle $ABC$ intersect at $M{}.$ Consider quadrilateral $ANMK$ and find the maximum possible number of its sides having length 1.
[i]Egor Bakaev[/i]
Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.
(a) In triangle $ABC$, angle $A$ is greater than angle $B$. Prove that the length of side $BC$ is greater than half the length of side $AB$.
(b) In the convex quadrilateral $ABCD$, the angle at $A$ is greater than the angle at $C$ and the angle at $D$ is greater than the angle at $B$. Prove that the length of side $BC$ is greater than half of the length of side $AD$.
(F Nazarov)
Consider an isosceles triangle $ABC$ with $AB = AC$. Let $\omega(XYZ)$ be the circumcircle of the triangle $XY Z$. The tangents to $\omega(ABC)$ through $B$ and $C$ meet at the point $D$. The point $F$ is marked on the arc $AB$ of $\omega(ABC)$ that does not contain $C$. Let $K$ be the point of intersection of lines $AF$ and $BD$ and $L$ the point of intersection of the lines $AB$ and $CF$. Let $T$ and $S$ be the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively. Suppose that the circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other at the point $P$. Prove that $P$ belongs to the line $AB$.
For a positive integer $n$, let $S(n) $be the set of complex numbers $z = x+iy$ ($x,y \in R$) with $ |z| = 1$ satisfying
$(x+iy)^n+(x-iy)^n = 2x^n$ .
(a) Determine $S(n)$ for $n = 2,3,4$.
(b) Find an upper bound (depending on $n$) of the number of elements of $S(n)$ for $n > 5$.
We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.
Let \( ABCDEF \) be a cyclic hexagon such that \( AD \parallel EF \). Points \( X \) and \( Y \) are marked on diagonals \( AE \) and \( DF \), respectively, such that \( CX = EX \) and \( BY = FY \). Let \( O \) be the intersection point of \( AE \) and \( FD \), \( P \) the intersection point of \( CX \) and \( BY \), and \( Q \) the intersection point of \( BF \) and \( CE \). Prove that points \( O, P, \) and \( Q \) are collinear.
[i]Proposed by Matthew Kurskyi[/i]
Given a non-decreasing unbounded sequence $a_n,$ construct a new sequence $b_n$ as follows
$$b_n = \frac{a_2 - a_1}{a_2} + \frac{a_3 - a_2}{a_3} + ... + \frac{a_n - a_{n-1}}{a_n}$$
Prove that $b_n$ is also unbounded.
Let $ABC$ be a triangle with $M, N, P$ as midpoints of the segments $BC, CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\angle BPM, \angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CN M, \angle MPN$. Denote $(\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\omega_2)$ as the circle of center $J$ and tangent to $MN$ at $E$.
a) Prove that $DE$ is parallel to $BC$.
b) Prove that the radical axis of two circles $(\omega_1), (\omega_2)$ bisects the segment $DE$.
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
[i]Proposed by El Salvador[/i]
The grid to the right consists of 74 unit squares, marked by gridlines. Partition the grid into five regions along the gridlines so that the areas of the regions are 1, 13, 19, 20, and 21. A square with a number should be contained in the region with that area.
[asy]unitsize(20);
path p = (5,0)--(3,0)--(3,1)--(1,1)--(1,2)--(0,2)--(0,7)--(1,7)--(1,8)--(3,8)--(3,9)--(5,9);
draw(p^^reflect((5,0),(5,3.14))*p);
int[] v = {0,1,1,0,0,0,0,0,1,1};
int[] h = {0,1,0,0,0,0,0,1,1};
for(int i=1; i<10; ++i) {
draw((i,v[i])--(i,9-v[i]),dotted);
}
for(int i=1; i<9; ++i) {
draw((h[i],i)--(10-h[i],i),dotted);
}
int[][] dord = {{1,4,21},{2,4,19},{3,4,1},{4,4,13},{5,4,20},{6,4,19},{7,4,20},{8,4,19},{2,2,21},{2,6,19},{4,7,19},{5,1,21},{7,6,13},{7,2,13}};
for(int i=0; i<14; ++i){
label(string(dord[i][2]),(dord[i][0]+.5,.5+dord[i][1]));
}
[/asy]
You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
While Peter was driving from home to work, he noticed that after driving 21 miles, the distance he had left to drive was 30 percent of the total distance from home to work. How many miles was his complete trip home to work?
There is a deck of cards placed at every points $A_1, A_2, \ldots , A_n$ and $O$, where $n \geq 3$. We can do one of the following two operations at each step:
$1)$ If there are more than 2 cards at some points $A_i$, we can withdraw three cards from that deck and place one each at $A_{i-1}, A_{i+1}$ and $O$. (Here $A_0=A_n$ and $A_{n+1}=A_1$);
$2)$ If there are more than or equal to $n$ cards at point $O$, we can withdraw $n$ cards from that deck and place one each at $A_1, A_2, \ldots , A_n$.
Show that if the total number of cards is more than or equal to $n^2+3n+1$, we can make the number of cards at every points more than or equal to $n+1$ after finitely many steps.
Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than
\[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\]
[i]Proposed by Soviet Union[/i]