Found problems: 1132
2007 JBMO Shortlist, 1
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.
2007 CentroAmerican, 3
Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$, $p$ and $q$, there exist not necessarily distinct integers $a \neq 0$, $b$, $c$ belonging to $S$, such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$. Determine the maximum number of elements that $S$ can have.
1995 IberoAmerican, 2
Let $n$ be a positive integer greater than 1. Determine all the collections of real numbers $x_1,\ x_2,\dots,\ x_n\geq1\mbox{ and }x_{n+1}\leq0$ such that the next two conditions hold:
(i) $x_1^{\frac12}+x_2^{\frac32}+\cdots+x_n^{n-\frac12}= nx_{n+1}^\frac12$
(ii) $\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1}$
1988 IMO Longlists, 36
[b]i.)[/b] Let $ABC$ be a triangle with $AB = 12$ and $AC = 16.$ Suppose $M$ is the midpoint of side $BC$ and points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, and suppose that lines $EF$ and $AM$ intersect at $G.$ If $AE = 2 \cdot AF$ then find the ratio
\[ \frac{EG}{GF} \]
[b]ii.)[/b] Let $E$ be a point external to a circle and suppose that two chords $EAB$ and $EDC$ meet at angle of $40^{\circ}.$ If $AB = BC = CD$ find the size of angle $ACD.$
1985 Dutch Mathematical Olympiad, 1
For some $ p$, the equation $ x^3 \plus{} px^2 \plus{} 3x \minus{} 10 \equal{} 0$ has three real solutions $ a,b,c$ such that $ c \minus{} b \equal{} b \minus{} a > 0$. Determine $ a,b,c,$ and $ p$.
2006 Greece Junior Math Olympiad, 4
If $x , y$ are real numbers such that $x^2 + xy + y^2 = 1$ , find the least and the greatest value( minimum and maximum) of the expression $K = x^3y + xy^3$
[u]Babis[/u]
[b] Sorry !!! I forgot to write that these 4 problems( 4 topics) were [u]JUNIOR LEVEL[/u][/b]
1989 IMO Longlists, 83
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]
2003 Federal Competition For Advanced Students, Part 1, 3
Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system
\[a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.\]
2007 AMC 12/AHSME, 9
A function $ f$ has the property that $ f(3x \minus{} 1) \equal{} x^{2} \plus{} x \plus{} 1$ for all real numbers $ x$. What is $ f(5)$?
$ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 31 \qquad \textbf{(D)}\ 111 \qquad \textbf{(E)}\ 211$
Kettering MO, 2005
Today was the 5th Kettering Olympiad - and here are the problems, which are very good intermediate problems.
1. Find all real $x$ so that $(1+x^2)(1+x^4)=4x^3$
2. Mark and John play a game. They have $100$ pebbles on a table. They take turns taking at least one at at most eight pebbles away. The person to claim the last pebble wins. Mark goes first. Can you find a way for Mark to always win? What about John?
3. Prove that
$\sin x + \sin 3x + \sin 5x + ... + \sin 11 x = (1-\cos 12 x)/(2 \sin x)$
4. Mark has $7$ pieces of paper. He takes some of them and splits each into $7$ pieces of paper. He repeats this process some number of times. He then tells John he has $2000$ pieces of paper. John tells him he is wrong. Why is John right?
5. In a triangle $ABC$, the altitude, angle bisector, and median split angle $A$ into four equal angles. Find the angles of $ABC.$
6. There are $100$ cities. There exist airlines connecting pairs of cities.
a) Find the minimal number of airlines such that with at most $k$ plane changes, one can go from any city to any other city.
b) Given that there are $4852$ airlines, show that, given any schematic, one can go from any city to any other city.
1993 All-Russian Olympiad, 4
On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?
1986 IMO Longlists, 26
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
1985 USAMO, 2
Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\] correct to four decimal places.
2013 Czech-Polish-Slovak Match, 3
For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2-rx$ and $x^3-rx$ are both rational, then $x$ is also rational.
[list](a) Prove the claim for $r \ge \frac43$ and $r \le 0$.
(b) Let $p,q$ be different odd primes such that $3p <4q$. Prove that the claim for $r=\frac{p}q$ does not hold.
[/list]
2010 AMC 10, 19
A circle with center $ O$ has area $ 156\pi$. Triangle $ ABC$ is equilateral, $ \overline{BC}$ is a chord on the circle, $ OA \equal{} 4\sqrt3$, and point $ O$ is outside $ \triangle ABC$. What is the side length of $ \triangle ABC$?
$ \textbf{(A)}\ 2\sqrt3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 4\sqrt3 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 18$
2012 Romania Team Selection Test, 1
Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.
1992 China Team Selection Test, 3
For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$
2004 France Team Selection Test, 1
If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$.
Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?
2008 AIME Problems, 11
In triangle $ ABC$, $ AB \equal{} AC \equal{} 100$, and $ BC \equal{} 56$. Circle $ P$ has radius $ 16$ and is tangent to $ \overline{AC}$ and $ \overline{BC}$. Circle $ Q$ is externally tangent to $ P$ and is tangent to $ \overline{AB}$ and $ \overline{BC}$. No point of circle $ Q$ lies outside of $ \triangle ABC$. The radius of circle $ Q$ can be expressed in the form $ m \minus{} n\sqrt {k}$, where $ m$, $ n$, and $ k$ are positive integers and $ k$ is the product of distinct primes. Find $ m \plus{} nk$.
2005 MOP Homework, 6
Solve the system of equations:
$x^2=\frac{1}{y}+\frac{1}{z}$,
$y^2=\frac{1}{z}+\frac{1}{x}$,
$z^2=\frac{1}{x}+\frac{1}{y}$.
in the real numbers.
2011-2012 SDML (High School), 1
The function $f$ is defined by $f\left(x\right)=x^2+3x$. Find the product of all solutions of the equation $f\left(2x-1\right)=6$.
1950 Miklós Schweitzer, 2
Consider three different planes and consider also one point on each of them. Give necessary and sufficient conditions for the existence of a quadratic which passes through the given points and whose tangent-plane at each of these points is the respective given plane.
2018 Indonesia Juniors, day 1
The problems are really difficult to find online, so here are the problems.
P1. It is known that two positive integers $m$ and $n$ satisfy $10n - 9m = 7$ dan $m \leq 2018$. The number $k = 20 - \frac{18m}{n}$ is a fraction in its simplest form.
a) Determine the smallest possible value of $k$.
b) If the denominator of the smallest value of $k$ is (equal to some number) $N$, determine all positive factors of $N$.
c) On taking one factor out of all the mentioned positive factors of $N$ above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.
I added this because my translation is a bit weird.
[hide=Indonesian Version] Diketahui dua bilangan bulat positif $m$ dan $n$ dengan $10n - 9m = 7$ dan $m \leq 2018$. Bilangan $k = 20 - \frac{18m}{n}$ merupakan suatu pecahan sederhana.
a) Tentukan bilangan $k$ terkecil yang mungkin.
b) Jika penyebut bilangan $k$ terkecil tersebut adalah $N$, tentukan semua faktor positif dari $N$.
c) Pada pengambilan satu faktor dari faktor-faktor positif $N$ di atas, tentukan peluang terambilnya satu faktor kelipatan 4.[/hide]
P2. Let the functions $f, g : \mathbb{R} \to \mathbb{R}$ be given in the following graphs.
[hide=Graph Construction Notes]I do not know asymptote, can you please help me draw the graphs? Here are its complete description:
For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with $X$ or $Y$ depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their $X$ or $Y$ coordinates only (since their other coordinates are definitely 0).
Graph (1) is the function $f$, who is a quadratic function with -2 and 4 as its $X$-intercepts and 4 as its $Y$-intercept. You also put $f$ right besides the curve you have, preferably just on the right-up direction of said curve.
Graph (2) is the function $g$, which is piecewise. For $x \geq 0$, $g(x) = \frac{1}{2}x - 2$, whereas for $x < 0$, $g(x) = - x - 2$. You also put $g$ right besides the curve you have, on the lower right of the line, on approximately $x = 2$.[/hide]
Define the function $g \circ f$ with $(g \circ f)(x) = g(f(x))$ for all $x \in D_f$ where $D_f$ is the domain of $f$.
a) Draw the graph of the function $g \circ f$.
b) Determine all values of $x$ so that $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$.
P3. The quadrilateral $ABCD$ has side lengths $AB = BC = 4\sqrt{3}$ cm and $CD = DA = 4$ cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral $ABCD$.
P4. There exists positive integers $x$ and $y$, with $x < 100$ and $y > 9$. It is known that $y = \frac{p}{777} x$, where $p$ is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of $y$.
P5. The 8-digit number $\overline{abcdefgh}$ (the original problem does not have an overline, which I fixed) is arranged from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Such number satisfies $a + c + e + g \geq b + d + f + h$. Determine the quantity of different possible (such) numbers.
1998 Vietnam Team Selection Test, 2
Let $d$ be a positive divisor of $5 + 1998^{1998}$. Prove that $d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2$, where $x, y$ are integers if and only if $d$ is congruent to 3 or 7 $\pmod{20}$.
2002 AMC 12/AHSME, 3
For how many positive integers $ n$ is $ n^2\minus{}3n\plus{}2$ a prime number?
$ \textbf{(A)}\ \text{none} \qquad
\textbf{(B)}\ \text{one} \qquad
\textbf{(C)}\ \text{two} \qquad
\textbf{(D)}\ \text{more than two, but finitely many}\\
\textbf{(E)}\ \text{infinitely many}$