Olympiad problems

2008 CentroAmerican, 5

Tags: polynomial , algebra

Find a polynomial $ p\left(x\right)$ with real coefficients such that $ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$ for all real $ x$ and $ p\left(1\right)\equal{}210$.

2006 Tournament of Towns, 2

Tags: incenter , perpendicular , geometry

The incircle of the quadrilateral $ABCD$ touches $AB,BC, CD$ and $DA$ at $E, F,G$ and $H$ respectively. Prove that the line joining the incentres of triangles $HAE$ and $FCG$ is perpendicular to the line joining the incentres of triangles $EBF$ and $GDH$. (4)

2016 BMT Spring, 16

Tags: geometry , 3d geometry , sphere , tetrahedron , analytic geometry , geometric inequality

What is the radius of the largest sphere that fits inside the tetrahedron whose vertices are the points $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$?

2017 Hanoi Open Mathematics Competitions, 3

Tags: diophantine equation , diophantine , number theory

The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is (A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.

2002 Hungary-Israel Binational, 3

Tags: number theory

Let $p \geq 5$ be a prime number. Prove that there exists a positive integer $a < p-1$ such that neither of $a^{p-1}-1$ and $(a+1)^{p-1}-1$ is divisible by $p^{2}$ .

2016 Postal Coaching, 4

Tags: number theory , diophantine equation , prime number

Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

2020 Thailand TSTST, 2

Tags: arithmetic function , sequence , number theory

For any positive integer $m \geq 2$, let $p(m)$ be the smallest prime dividing $m$ and $P(m)$ be the largest prime dividing $m$. Let $C$ be a positive integer. Define sequences $\{a_n\}$ and $\{b_n\}$ by $a_0 = b_0 = C$ and, for each positive integer $k$ such that $a_{k-1}\geq 2$, $$a_k=a_{k-1}-\frac{a_{k-1}}{p(a_{k-1})};$$ and, for each positive integer $k$ such that $b_{k-1}\geq 2$, $$b_k=b_{k-1}-\frac{b_{k-1}}{P(b_{k-1})}$$ It is easy to see that both $\{a_n\}$ and $\{b_n\}$ are finite sequences which terminate when they reach the number $1$. Prove that the numbers of terms in the two sequences are always equal.

2021 Cyprus JBMO TST, 4

Tags: geometry

Let $\triangle AB\varGamma$ be an acute-angled triangle with $AB < A\varGamma$, and let $O$ be the center of the circumcircle of the triangle. On the sides $AB$ and $A \varGamma$ we select points $T$ and $P$ respectively such that $OT=OP$. Let $M,K$ and $\varLambda$ be the midpoints of $PT,PB$ and $\varGamma T$ respectively. Prove that $\angle TMK = \angle M\varLambda K$.

2005 Tournament of Towns, 3

Tags: logic

John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kopeks wins. Which player has a winning strategy? [i](5 points)[/i]

2007 Turkey MO (2nd round), 1

Tags: circumcircle , geometry

In an acute triangle $ABC$, the circle with diameter $AC$ intersects $AB$ and $AC$ at $K$ and $L$ different from $A$ and $C$ respectively. The circumcircle of $ABC$ intersects the line $CK$ at the point $F$ different from $C$ and the line $AL$ at the point $D$ different from $A$. A point $E$ is choosen on the smaller arc of $AC$ of the circumcircle of $ABC$ . Let $N$ be the intersection of the lines $BE$ and $AC$ . If $AF^{2}+BD^{2}+CE^{2}=AE^{2}+CD^{2}+BF^{2}$ prove that $\angle KNB= \angle BNL$ .

2016 Bangladesh Mathematical Olympiad, 3

Tags: geometry , angle chasing

$\triangle ABC$ is isosceles $AB = AC$. $P$ is a point inside $\triangle ABC$ such that $\angle BCP = 30$ and $\angle APB = 150$ and $\angle CAP = 39$. Find $\angle BAP$.

1954 AMC 12/AHSME, 18

Tags: inequalities

Of the following sets, the one that includes all values of $ x$ which will satisfy $ 2x \minus{} 3 > 7 \minus{} x$ is: $ \textbf{(A)}\ x > 4 \qquad \textbf{(B)}\ x < \frac {10}{3} \qquad \textbf{(C)}\ x \equal{} \frac {10}{3} \qquad \textbf{(D)}\ x > \frac {10}{3} \qquad \textbf{(E)}\ x < 0$

2008 Irish Math Olympiad, 1

Tags: geometry , number theory

Find, with proof, all triples of integers $ (a,b,c)$ such that $ a, b$ and $ c$ are the lengths of the sides of a right angled triangle whose area is $ a \plus{} b \plus{} c$

2002 Singapore MO Open, 4

Tags: functional equation , functional , algebra

Find all real-valued functions $f : Q \to R$ defined on the set of all rational numbers $Q$ satisfying the conditions $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y$ in $Q$ and $f(1) = 2002.$ Justify your answers.

1967 IMO Longlists, 10

Tags: perimeter , square , dissection , triangulation , geometry

The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.

1995 AMC 12/AHSME, 21

Tags: geometry , rectangle , analytic geometry , graphing lines , slope , circumcircle

Two nonadjacent vertices of a rectangle are $(4,3)$ and $(-4,-3)$, and the coordinates of the other two vertices are integers. The number of such rectangles is $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

1975 AMC 12/AHSME, 29

Tags:

What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$? $ \textbf{(A)}\ 972 \qquad\textbf{(B)}\ 971 \qquad\textbf{(C)}\ 970 \qquad\textbf{(D)}\ 969 \qquad\textbf{(E)}\ 968 $

2015 Brazil National Olympiad, 4

Tags: number theory

Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors. a) Prove that $$d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1$$ iff $n$ is prime or $n=4$. b) Determine the three positive integers such that $$d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.$$

2014 Israel National Olympiad, 7

Tags: algebra , polynomial , root

Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$.

2007 USA Team Selection Test, 1

Tags: asymptote , geometric transformation , geometry

Circles $ \omega_1$ and $ \omega_2$ meet at $ P$ and $ Q$. Segments $ AC$ and $ BD$ are chords of $ \omega_1$ and $ \omega_2$ respectively, such that segment $ AB$ and ray $ CD$ meet at $ P$. Ray $ BD$ and segment $ AC$ meet at $ X$. Point $ Y$ lies on $ \omega_1$ such that $ PY \parallel BD$. Point $ Z$ lies on $ \omega_2$ such that $ PZ \parallel AC$. Prove that points $ Q,X,Y,Z$ are collinear.

2001 Moldova Team Selection Test, 7

Tags: polynomial

Let $(P_n(X))_{n\in\mathbb{N}}$ be a sequence of polynomials defined as: $P_1(X)=X-1, P_2(X)=X^2-X-1, P_n(X)=XP_{n-1}(X)-P_{n-2}(X), \forall n>2$. For every nonnegative integer $n{}$ find all roots of the polynomial $P_n(X)$.

2009 IMC, 2

Tags:

Let $A,B,C$ be real square matrices of the same order, and suppose $A$ is invertible. Prove that \[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]

1994 Abels Math Contest (Norwegian MO), 3a

Tags: inequalities , algebra , product

Let $x_1,x_2,...,x_{1994}$ be positive real numbers. Prove that $$\left(\frac{x_1}{x_2}\right)^{\frac{x_1}{x_2}}\left(\frac{x_2}{x_3}\right)^{\frac{x_2}{x_3}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1993}}{x_{1994}}} \ge \left(\frac{x_1}{x_2}\right)^{\frac{x_2}{x_1}}\left(\frac{x_2}{x_3}\right)^{\frac{x_3}{x_2}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1994}}{x_{1993}}}$$

1959 Miklós Schweitzer, 10

Tags:

[b]10.[/b] Prove that if a graph with $2n+1$ vertices has at least $3n+1$ edges, then the graph contains a circuit having an even number of edges. Prove further that this statemente does not hold for $3n$ edges. (By a circuit, we mean a closed line which does not intersect itself.) [b](C. 5)[/b]

2008 Putnam, A5

Tags: algebra , polynomial , rotation , trigonometry , analytic geometry

Let $ n\ge 3$ be an integer. Let $ f(x)$ and $ g(x)$ be polynomials with real coefficients such that the points $ (f(1),g(1)),(f(2),g(2)),\dots,(f(n),g(n))$ in $ \mathbb{R}^2$ are the vertices of a regular $ n$-gon in counterclockwise order. Prove that at least one of $ f(x)$ and $ g(x)$ has degree greater than or equal to $ n\minus{}1.$