Olympiad problems

2015 BMT Spring, 11

Tags: algebra

Let $r, s$, and $t$ be the three roots of the equation $8x^3 + 1001x + 2008 = 0$. Find $(r + s)^3 + (s + t)^3 + (t + r)^3$ .

2014 Contests, 1

Tags: geometry , circumcircle , geometry proposed , easy

Points $M$, $N$, $K$ lie on the sides $BC$, $CA$, $AB$ of a triangle $ABC$, respectively, and are different from its vertices. The triangle $MNK$ is called[i] beautiful[/i] if $\angle BAC=\angle KMN$ and $\angle ABC=\angle KNM$. If in the triangle $ABC$ there are two beautiful triangles with a common vertex, prove that the triangle $ABC$ is right-angled. [i]Proposed by Nairi M. Sedrakyan, Armenia[/i]

1974 Dutch Mathematical Olympiad, 4

Tags: Perfect Square , number theory

For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

2019 Junior Balkan MO, 2

Tags: algebra , Junior , JBMO , 2019 , Balkan , inequalities

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

2014 May Olympiad, 4

Tags: geometry , right triangle , Isosceles Triangle , Equilateral Triangle , angle

Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$

1979 USAMO, 1

Tags: AMC , USA(J)MO , USAMO , calculus , integration , modular arithmetic

Determine all non-negative integral solutions $ (n_{1},n_{2},\dots , n_{14}) $ if any, apart from permutations, of the Diophantine Equation \[n_{1}^{4}+n_{2}^{4}+\cdots+n_{14}^{4}=1,599.\]

2012 Princeton University Math Competition, B2

Tags: number theory , PuMAC

Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers. Find $\Sigma_{k=1}^{n}(p_k + a_k)$

2007 Iran MO (3rd Round), 5

Tags: algebra , polynomial , geometry , function , algebra proposed

Prove that for two non-zero polynomials $ f(x,y),g(x,y)$ with real coefficients the system: \[ \left\{\begin{array}{c}f(x,y)\equal{}0\\ g(x,y)\equal{}0\end{array}\right.\] has finitely many solutions in $ \mathbb C^{2}$ if and only if $ f(x,y)$ and $ g(x,y)$ are coprime.

2008 China Team Selection Test, 2

Tags: modular arithmetic , number theory , relatively prime , number theory proposed

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

2014 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , equal angles , midpoints , diameter

Let $ABC$ be an acute triangle and let $O$ be its circumcentre. Now, let the diameter $PQ$ of circle $ABC$ intersects sides $AB$ and $AC$ in their interior at$ D$ and $E$, respectively. Now, let $F$ and $G$ be the midpoints of $CD$ and $BE$. Prove that $\angle FOG=\angle BAC$

2008 Moldova Team Selection Test, 3

Tags: trigonometry , geometry proposed , geometry

In triangle $ ABC$ the bisector of $ \angle ACB$ intersects $ AB$ at $ D$. Consider an arbitrary circle $ O$ passing through $ C$ and $ D$, so that it is not tangent to $ BC$ or $ CA$. Let $ O\cap BC \equal{} \{M\}$ and $ O\cap CA \equal{} \{N\}$. a) Prove that there is a circle $ S$ so that $ DM$ and $ DN$ are tangent to $ S$ in $ M$ and $ N$, respectively. b) Circle $ S$ intersects lines $ BC$ and $ CA$ in $ P$ and $ Q$ respectively. Prove that the lengths of $ MP$ and $ NQ$ do not depend on the choice of circle $ O$.

2017 Serbia Team Selection Test, 1

Tags: inequalities , geometry

Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. Define points $E$ and $F$ on $AC$ and $B$, respectively, such that $DE=DF$ and $\angle EDF =\angle BAC$. Prove that $$DE\geq \frac {AB+AC} 4.$$

1984 National High School Mathematics League, 4

Tags:

The number of real roots of the equation $\sin x=\lg x$ is $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}$more than $3$

2010 Contests, 3

Tags: geometry , power of a point , radical axis , geometry proposed

Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.

2007 Abels Math Contest (Norwegian MO) Final, 2

Tags: ratio , Cyclic , pentagon , circumradius , circles , Regular , geometry

The vertices of a convex pentagon $ABCDE$ lie on a circle $\gamma_1$. The diagonals $AC , CE, EB, BD$, and $DA$ are tangents to another circle $\gamma_2$ with the same centre as $\gamma_1$. (a) Show that all angles of the pentagon $ABCDE$ have the same size and that all edges of the pentagon have the same length. (b) What is the ratio of the radii of the circles $\gamma_1$ and $\gamma_2$? (The answer should be given in terms of integers, the four basic arithmetic operations and extraction of roots only.)

1990 AMC 12/AHSME, 8

Tags: AMC

The number of real solutions of the equation \[|x-2|+|x-3|=1\] is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }\text{more than }3\qquad$

1987 IMO Longlists, 3

Tags: combinatorics proposed , combinatorics

A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop $X$ to bus stop $Y \neq X$, then we shall say [i]$Y$ can be reached from $X$[/i]. We shall use the phrase [i]$Y$ comes after $X$[/i] when we wish to express that every bus stop from which the bus stop $X$ can be reached is a bus stop from which the bus stop $Y$ can be reached, and every bus stop that can be reached from $Y$ can also be reached from $X$. A visitor to this town discovers that if $X$ and $Y$ are any two different bus stops, then the two sentences [i]“$Y$ can be reached from $X$”[/i] and [i]“$Y$ comes after $X$”[/i] have exactly the same meaning in this town. Let $A$ and $B$ be two bus stops. Show that of the following two statements, exactly one is true: [b] (i)[/b] $B$ can be reached from $A;$ [b] (ii) [/b] $A$ can be reached from $B.$

1953 Moscow Mathematical Olympiad, 239

Tags: Locus , analytic geometry , trigonometry

On the plane find the locus of points whose coordinates satisfy $sin(x + y) = 0$.

2012 Today's Calculation Of Integral, 857

Tags: calculus , integration , limit , trigonometry , geometry , geometric transformation , rotation

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

2022 Princeton University Math Competition, A2 / B4

Tags: number theory

Compute the sum of all positive integers whose positive divisors sum to $186.$

2012 NZMOC Camp Selection Problems, 5

Tags: geometry , angle bisector , parallel , equal angles

Let $ABCD$ be a quadrilateral in which every angle is smaller than $180^o$. If the bisectors of angles $\angle DAB$ and $\angle DCB$ are parallel, prove that $\angle ADC = \angle ABC$

2004 Alexandru Myller, 3

Tags: abstract algebra , Ring Theory , nilpotence , superior algebra

Prove that the number of nilpotent elements of a commutative ring with an order greater than $ 8 $ and congruent to $ 3 $ modulo $ 6 $ is at most a third of the order of the ring.

1975 Czech and Slovak Olympiad III A, 6

Tags: algebra , linear algebra , Subsets

Let $\mathbf M\subseteq\mathbb R^2$ be a set with the following properties: 1) there is a pair $(a,b)\in\mathbf M$ such that $ab(a-b)\neq0,$ 2) if $\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M$ and $c\in\mathbb R$ then also \[\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M.\] Show that in fact \[\mathbf M=\mathbb R^2.\]

2011 Tournament of Towns, 2

Tags: geometry , rectangle , perimeter , Integer , combinatorial geometry

A rectangle is divided by $10$ horizontal and $10$ vertical lines into $121$ rectangular cells. If $111$ of them have integer perimeters, prove that they all have integer perimeters.

2022 Germany Team Selection Test, 2

Tags: geometry , circumcircle , projective geometry

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.