Found problems: 85335
2014 AMC 12/AHSME, 13
Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1,a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$?
${ \textbf{(A)}\ \dfrac{3+\sqrt{3}}{2}\qquad\textbf{(B)}\ \dfrac52\qquad\textbf{(C)}\ \dfrac{3+\sqrt{5}}{2}\qquad\textbf{(D)}}\ \dfrac{3+\sqrt{6}}{2}\qquad\textbf{(E)}\ 3 $
1989 AMC 12/AHSME, 29
Find $\displaystyle \sum_{k=0}^{49}(-1)^k\binom{99}{2k}$, where $\binom{n}{j}=\frac{n!}{j!(n-j)!}$.
$ \textbf{(A)}\ -2^{50} \qquad\textbf{(B)}\ -2^{49} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2^{49} \qquad\textbf{(E)}\ 2^{50} $
1960 IMO Shortlist, 5
Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$).
a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$;
b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.
2005 AMC 12/AHSME, 12
A line passes through $ A(1,1)$ and $ B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $ A$ and $ B$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$
2023 Turkey Team Selection Test, 9
The points $ A,B,K,L,X$ lies of the circle $\Gamma$ in that order such that the arcs $\widehat{BK}$ and $\widehat{KL}$ are equal. The circle that passes through $A$ and tangent to $BK$ at $B$ intersects the line segment $KX$ at $P$ and $Q$. The circle that passes through $A$ and tangent to $BL$ at $B$ intersect the line segment $BX$ for the second time at $T$. Prove that $\angle{PTB} = \angle{XTQ}$
1976 IMO Longlists, 24
Let $0 \le x_1 \le x_2\le\cdots\le x_n \le 1$. Prove that for all $A \ge 1$, there exists an interval $I$ of length $2\sqrt[n]{A}$ such that for all $x \in I$,
\[|(x - x_1)(x - x_2) \cdots (x -x_n)| \le A.\]
2013 VTRMC, Problem 3
Define a sequence $(a_n)$ for $n\ge1$ by $a_1=2$ and $a_{n+1}=a_n^{1+n^{-3/2}}$. Is $(a_n)$ convergent (i.e. $\lim_{n\to\infty}a_n<\infty$)?
1986 IMO Longlists, 1
Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$. Prove the inequality
\[1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k\]
for all real $b \geq 0.$
2015 ASDAN Math Tournament, 6
Let $f(x)=x^4-4x^3-3x^2-4x+1$. Compute the sum of the real roots of $f(x)$.
2025 Taiwan TST Round 1, 5
A country has 2025 cites, with some pairs of cities having bidirectional flight routes between them. For any pair of the cities, the flight route between them must be operated by one of the companies $X, Y$ or $Z$. To avoid unfairly favoring specific company, the regulation ensures that if there have three cities $A, B$ and $C$, with flight routes $A \leftrightarrow B$ and $A \leftrightarrow C$ operated by two different companies, then there must exist flight route $B \leftrightarrow C$ operated by the third company different from $A \leftrightarrow B$ and $A \leftrightarrow C$ .
Let $n_X$, $n_Y$ and $n_Z$ denote the number of flight routes operated by companies $X, Y$ and $Z$, respectively. It is known that, starting from a city, we can arrive any other city through a series of flight routes (not necessary operated by the same company). Find the minimum possible value of $\max(n_X, n_Y , n_Z)$.
[i]
Proposed by usjl and YaWNeeT[/i]
2014 Tajikistan Team Selection Test, 3
Let $a$, $b$, $c$ be side length of a triangle. Prove the inequality
\begin{align*} \sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ca + a^2} \leq \sqrt{5a^2 + 5b^2 + 5c^2 + 4ab + 4 bc + 4ca}.\end{align*}
1988 Polish MO Finals, 2
The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = a_2 = a_3 = 1$, $a_{n+3} = a_{n+2}a_{n+1} + a_n$. Show that for any positive integer $r$ we can find $s$ such that $a_s$ is a multiple of $r$.
1969 IMO Shortlist, 56
Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$
2019 Online Math Open Problems, 21
Let $p$ and $q$ be prime numbers such that $(p-1)^{q-1}-1$ is a positive integer that divides $(2q)^{2p}-1$. Compute the sum of all possible values of $pq$.
[i]Proposed by Ankit Bisain[/i]
2009 Germany Team Selection Test, 1
Let $ ABCD$ be a chordal/cyclic quadrilateral. Consider points $ P,Q$ on $ AB$ and $ R,S$ on $ CD$ with
\[ \overline{AP}: \overline{PB} \equal{} \overline{CS}: \overline{SD}, \quad \overline{AQ}: \overline{QB} \equal{} \overline{CR}: \overline{RD}.\]
How to choose $ P,Q,R,S$ such that $ \overline{PR} \cdot \overline{AB} \plus{} \overline{QS} \cdot \overline{CD}$ is minimal?
1936 Moscow Mathematical Olympiad, 022
Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.
2006 IMO Shortlist, 1
Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]
Estonia Open Senior - geometry, 1994.2.2
The two sides $BC$ and $CD$ of an inscribed quadrangle $ABCD$ are of equal length. Prove that the area of this quadrangle is equal to $S =\frac12 \cdot AC^2 \cdot \sin \angle A$
2012 Junior Balkan Team Selection Tests - Moldova, 1
Let $ 1\leq a,b,c,d,e,f,g,h,k \leq 9 $ and $ a,b,c,d,e,f,g,h,k $ are different integers, find the minimum value of the expression $ E = a*b*c+d*e*f+g*h*k $ and prove that it is minimum.
2022 USAMO, 1
Let $a$ and $b$ be positive integers. The cells of an $(a+b+1)\times (a+b+1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.
1964 Spain Mathematical Olympiad, 8
The points $A$ and $B$ lie on a horizontal line over a vertical plane. We consider the semicircumference passing through $A$ and $B$ that lies under the horizontal line. A segment of length $a$, with the same diameter that the semicircumference, moves in a way that always contains the point $A$ and one of its extremes lies always on the semicircumference. Determine the value of the cosine of the angle between this segment and the horizontal line that makes the medium point of the segment to be as down as possible.
2017 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$.
[i]Proposed by Vincent Huang
1999 Turkey Team Selection Test, 2
Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.
2019 Federal Competition For Advanced Students, P2, 1
Determine all functions $f: R\to R$, such that $f (2x + f (y)) = x + y + f (x)$ for all $x, y \in R$.
(Gerhard Kirchner)
2002 Tournament Of Towns, 5
[list]
[*] There are $128$ coins of two different weights, $64$ each. How can one always find two coins of different weights by performing no more than $7$ weightings on a regular balance?
[*] There are $8$ coins of two different weights, $4$ each. How can one always find two coins of different weights by performing two weightings on a regular balance?[/list]