Found problems: 85335
1988 Swedish Mathematical Competition, 3
Show that if $x_1+x_2+x_3 = 0$ for real numbers $x_1,x_2,x_3$, then $x_1x_2+x_2x_3+x_3x_1\le 0$.
Find all $n \ge 4$ for which $x_1+x_2+...+x_n = 0$ implies $x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0$.
2022 AMC 10, 2
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?
[asy]
import olympiad;
size(180);
real r = 3, s = 5, t = sqrt(r*r+s*s);
defaultpen(linewidth(0.6) + fontsize(10));
pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D));
label("$A$",A,SW);
label("$B$", B, NW);
label("$C$",C,NE);
label("$D$",D,SE);
label("$P$",P,S);
[/asy]
$\textbf{(A) }3\sqrt 5 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }6\sqrt 5 \qquad
\textbf{(D) }20\qquad
\textbf{(E) }25$
2012 Gheorghe Vranceanu, 2
Calculate the limit of the following sequences:
[b]a)[/b] n^{n!}/(n!)^n
[b]b)[/b] n^{ln n}/n!
[i]Adrian Troie[/i]
2020 Argentina National Olympiad, 3
Let $ABC$ be a right isosceles triangle with right angle at $A$. Let $E$ and $F$ be points on A$B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$. Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$. Calculate the measure of angle $\angle FDC$.
2005 Dutch Mathematical Olympiad, 3
Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?
2023 Switzerland - Final Round, 1
Let $ABC$ be an acute triangle with incenter $I$. On its circumcircle, let $M_A$, $M_B$ and $M_C$ be the midpoints of minor arcs $BC, CA$ and $AB$, respectively. Prove that the reflection $M_A$ over the line $IM_B$ lies on the circumcircle of the triangle $IM_BM_C$.
2010 Czech-Polish-Slovak Match, 1
Given any collection of $2010$ nondegenerate triangles, their sides are painted so that each triangle has one red side, one blue side, and one white side. For each color, arrange the side lengths in order: [list][*]let $b_1\le b_2\le\cdots\le b_{2011}$ denote the lengths of the blue sides;
[*]let $r_1\le r_2\le\cdots\le r_{2011}$ denote the lengths of the red sides; and
[*]let $w_1\le w_2\le\cdots\le w_{2011}$ denote the lengths of the white sides.[/list] Find the largest integer $k$ for which there necessarily exists at least $k$ indices $j$ such that $b_j$, $r_j$, $w_j$ are the side lengths of a nondegenerate triangle.
2019 Estonia Team Selection Test, 11
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.
2003 German National Olympiad, 4
From the midpoints of the sides of an acute-angled triangle, perpendiculars are drawn to the adjacent sides. The resulting six straight lines bound the hexagon. Prove that its area is half the area of the original triangle.
2013 Online Math Open Problems, 19
$A,B,C$ are points in the plane such that $\angle ABC=90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA=20$ and $BC=21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]Ray Li[/i]
2009 Stanford Mathematics Tournament, 4
How many ways are there to write $657$ as a sum of powers of two where each power of two is used at
most twice in the sum? For example, $256+256+128+16+1$ is a valid sum.
2022 USEMO, 4
Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points $P, Q, R, S$ lie in the interiors of segments $AB, BC, CD, DA,$ respectively, such that $$\angle PDA = \angle PCB, \text{ } \angle QAB = \angle QDC, \text{ } \angle RBC = \angle RAD, \text{ and } \angle SCD = \angle SBA.$$ Let $AQ$ intersect $BS$ at $X$, and $DQ$ intersect $CS$ at $Y$. Prove that lines $PR$ and $XY$ are either parallel or coincide.
[i]Tilek Askerbekov[/i]
2019 Belarusian National Olympiad, 11.1
[b]a)[/b] Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points.
[b]b)[/b] Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values.
[i](I. Gorodnin)[/i]
2015 USA TSTST, 2
Let ABC be a scalene triangle. Let $K_a$, $L_a$ and $M_a$ be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time at point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC.
(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)
[i]Proposed by Ivan Borsenco[/i]
2024 Spain Mathematical Olympiad, 4
Let $a,b,c,d$ be real numbers satisfying \[abcd=1\quad \text{and}\quad a+\frac1a+b+\frac1b+c+\frac1c+d+\frac1d=0.\] Prove that at least one of the numbers $ab$, $ac$, $ad$ equals $-1$.
2020 LIMIT Category 2, 1
Find the number of $f:\{1,\ldots, 5\}\to \{1,\ldots, 5\}$ such that $f(f(x))=x$
(A)$26$
(B)$41$
(C)$120$
(D)$60$
2007 Mongolian Mathematical Olympiad, Problem 4
If $x,y,z\in\mathbb N$ and $xy=z^2+1$ prove that there exists integers $a,b,c,d$ such that $x=a^2+b^2$, $y=c^2+d^2$, $z=ac+bd$.
1994 Hungary-Israel Binational, 1
Let $ m$ and $ n$ be two distinct positive integers. Prove that there exists a real number $ x$ such that $ \frac {1}{3}\le\{xn\}\le\frac {2}{3}$ and $ \frac {1}{3}\le\{xm\}\le\frac {2}{3}$. Here, for any real number $ y$, $ \{y\}$ denotes the fractional part of $ y$. For example $ \{3.1415\} \equal{} 0.1415$.
2024 Taiwan TST Round 3, G
Let $ABC$ be a triangle such that the angular bisector of $\angle BAC$, the $B$-median and the perpendicular bisector of $AB$ intersect at a single point $X$. Let $H$ be the orthocenter of $ABC$. Show that $\angle BXH = 90^{\circ}$.
[i]Proposed by usjl[/i]
2017 Pan-African Shortlist, A6
Let $n \geq 1$ be an integer, and $a_0, a_1, \dots, a_{n-1}$ be real numbers such that
\[
1 \geq a_{n-1} \geq a_{n-2} \geq \dots \geq a_1 \geq a_0 \geq 0.
\]
We assume that $\lambda$ is a real root of the polynomial
\[
x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0.
\]
Prove that $|\lambda| \leq 1$.
2016 Oral Moscow Geometry Olympiad, 5
From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).
2011 ELMO Shortlist, 1
Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that
a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$;
b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$.
Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if
\[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\]
[i]Evan O'Dorney.[/i]
1956 AMC 12/AHSME, 38
In a right triangle with sides $ a$ and $ b$, and hypotenuse $ c$, the altitude drawn on the hypotenuse is $ x$. Then:
$ \textbf{(A)}\ ab \equal{} x^2 \qquad\textbf{(B)}\ \frac {1}{a} \plus{} \frac {1}{b} \equal{} \frac {1}{x} \qquad\textbf{(C)}\ a^2 \plus{} b^2 \equal{} 2x^2$
$ \textbf{(D)}\ \frac {1}{x^2} \equal{} \frac {1}{a^2} \plus{} \frac {1}{b^2} \qquad\textbf{(E)}\ \frac {1}{x} \equal{} \frac {b}{a}$
2014 Harvard-MIT Mathematics Tournament, 8
The numbers $2^0, 2^1, \dots , 2{}^1{}^5, 2{}^1{}^6 = 65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?
2019 Saint Petersburg Mathematical Olympiad, 4
Given a convex quadrilateral $ABCD$. The medians of the triangle $ABC$ intersect at point $M$, and the medians of the triangle $ACD$ at point$ N$. The circle, circumscibed around the triangle $ACM$, intersects the segment $BD$ at the point $K$ lying inside the triangle $AMB$ . It is known that $\angle MAN = \angle ANC = 90^o$. Prove that $\angle AKD = \angle MKC$.