Found problems: 85335
2024 Ukraine National Mathematical Olympiad, Problem 5
Real numbers $a, b, c$ are such that
$$a^2+c-bc = b^2+a-ca = c^2+b-ab$$
Does it follow that $a=b=c$?
[i]Proposed by Mykhailo Shtandenko[/i]
1981 AMC 12/AHSME, 30
If $ a$, $ b$, $ c$, and $ d$ are the solutions of the equation $ x^4 \minus{} bx \minus{} 3 \equal{} 0$, then an equation whose solutions are
\[ \frac {a \plus{} b \plus{} c}{d^2}, \frac {a \plus{} b \plus{} d}{c^2}, \frac {a \plus{} c \plus{} d}{b^2}, \frac {b \plus{} c \plus{} d}{a^2}
\]is
$ \textbf{(A)}\ 3x^4 \plus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(B)}\ 3x^4 \minus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(C)}\ 3x^4 \plus{} bx^3 \minus{} 1 \equal{} 0$
$ \textbf{(D)}\ 3x^4 \minus{} bx^3 \minus{} 1 \equal{} 0\qquad \textbf{(E)}\ \text{none of these}$
1993 AIME Problems, 14
A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.
2021 BMT, 1
Let $x$ be a real number such that $x^2 -x+1 = 7$ and $x^2 +x+1 = 13$. Compute the value of $x^4$.
2020 Regional Olympiad of Mexico Southeast, 5
Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.
2017 Serbia Team Selection Test, 2
Initally a pair $(x, y)$ is written on the board, such that exactly one of it's coordinates is odd. On such a pair we perform an operation to get pair $(\frac x 2, y+\frac x 2)$ if $2|x$ and $(x+\frac y 2, \frac y 2)$ if $2|y$. Prove that for every odd $n>1$ there is a even positive integer $b<n$ such that starting from the pair $(n, b)$ we will get the pair $(b, n)$ after finitely many operations.
1984 All Soviet Union Mathematical Olympiad, 388
The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $$|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$$
2016 CCA Math Bonanza, L5.1
The first question was asked in Set 4. The second question was asked in Set 5.
Question) Eshaan the Elephant has a long memory. He remembers that out of the integers $0, 1, 2, \dots, 15$, one of them is special. Submit to the grader an ordered 4-tuple of subsets of $0, 1, 2, \dots, 15$ and they will tell you whether the special number is in each. You can then submit your guess for the special number on the next round for points. (You might want to write down a copy of your submission somewhere other than your answer sheet. Note that this question itself is not worth any points, though the corresponding problem in Set 5 is.)
Question) Eshaan the Elephant has a long memory. He remembers that out of the integers $0, 1, 2, \dots, 15$, one of them is special. You have submitted an ordered 4-tuple of subsets of $0, 1, 2, \dots, 15$. Here is your reply from the grader.
\begin{tabular}{|c|c|c|c|}
\hline
1 & 2 & 3 & 4 \\ \hline
Y/N & Y/N & Y/N & Y/N \\ \hline
\end{tabular}
What is the special number?
[i]2016 CCA Math Bonanza Lightning #5.1[/i]
2005 Gheorghe Vranceanu, 2
Let be a natural number $ n\ge 2 $ and a real number $ r>1. $ Determine the natural numbers $ k $ having the property that the affixes of $ r^ke^{\pi ki/n} ,r^{k+1}e^{\pi (k+1)i/n} ,r^{k+n}e^{\pi (k+n)i/n} ,r^{k+n+1}e^{\pi (k+n+1) i/n} $ in the complex plane represent the vertices of a trapezoid.
2025 India National Olympiad, P6
Let $b \geqslant 2$ be a positive integer. Anu has an infinite collection of notes with exactly $b-1$ copies of a note worth $b^k-1$ rupees, for every integer $k\geqslant 1$. A positive integer $n$ is called payable if Anu can pay exactly $n^2+1$ rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers.
[i]Proposed by Shantanu Nene[/i]
2011 China National Olympiad, 3
Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that
[b](a)[/b] The sum of the elements of $A$ is $0,$
[b](b)[/b] For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$.
Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that
\[|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.\]
Where $|X|$ denote the numbers of the elements in set $X$.
1990 Tournament Of Towns, (263) 1
Suppose two positive real numbers are given. Prove that if their sum is less than their product then their sum is greater than four.
(N Vasiliev, Moscow)
2024 Chile National Olympiad., 1
Let \( f(x) = \frac{100^x}{100^x + 10} \). Determine the value of:
\[
f\left( \frac{1}{2024} \right) - f\left( \frac{2}{2024} \right) + f\left( \frac{3}{2024} \right) - f\left( \frac{4}{2024} \right) + \ldots - f\left( \frac{2022}{2024} \right) + f\left( \frac{2023}{2024} \right)
\]
2024 Singapore Senior Math Olympiad, Q5
Let $a_1,a_2,\dots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$, $$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.
2017 Czech-Polish-Slovak Junior Match, 3
Prove that for all real numbers $x, y$ holds $(x^2 + 1)(y^2 + 1) \ge 2(xy - 1)(x + y)$.
For which integers $x, y$ does equality occur?
2010 Contests, 4
How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for $ j \equal{} 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?
2006 Greece JBMO TST, 3
Find the angle $\angle A$ of a triangle $ABC$, when we know it's altitudes $BD$ and $CE$ intersect in an interior point $H$ of the triangle and $BH=2HD$ and $CH=HE$.
2017-2018 SDPC, 3
Let $n > 2$ be a fixed positive integer. For a set $S$ of $n$ points in the plane, let $P(S)$ be the set of perpendicular bisectors of pairs of distinct points in $S$. Call set $S$ [i]complete[/i] if no two (distinct) pairs of points share the same perpendicular bisector, and every pair of lines in $P(S)$ intersects. Let $f(S)$ be the number of distinct intersection points of pairs of lines in $P(S)$.
(a) Find all complete sets $S$ such that $f(S) = 1$.
(b) Let $S$ be a complete set with $n$ points. Show that if $f(S)>1$, then $f(S) \geq n$.
2002 Iran Team Selection Test, 2
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
2015 AMC 12/AHSME, 21
A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?
$\textbf{(A) }5\sqrt2+4\qquad\textbf{(B) }\sqrt{17}+7\qquad\textbf{(C) }6\sqrt2+3\qquad\textbf{(D) }\sqrt{15}+8\qquad\textbf{(E) }12$
2022 ELMO Revenge, 1
In terms of $p$ and $k$, compute the number of solutions in positive integers to the equation
$ab+bc+ca=p^{2k}$ satisfying $a\leq b\leq c$ where $p$ is a fixed prime and $k$ is a fixed positive integer.
[i]Proposed by Alexander Wang[/i]
2001 Federal Competition For Advanced Students, Part 2, 3
Let be given a semicircle with the diameter $AB$, and points $C,D$ on it such that $AC = CD$. The tangent at $C$ intersects the line $BD$ at $E$. The line $AE$ intersects the arc of the semicircle at $F$. Prove that $CF < FD$.
2008 China Team Selection Test, 3
Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.
1988 IMO Shortlist, 25
A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.
2011 Math Prize For Girls Problems, 14
If $0 \le p \le 1$ and $0 \le q \le 1$, define $F(p, q)$ by
\[
F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).
\]
Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $G(p)$?