Olympiad problems

2002 Bundeswettbewerb Mathematik, 1

Tags: induction , combinatorics

A pile of cards, numbered with $1$, $2$, ..., $n$, is being shuffled. Afterwards, the following operation is repeatedly performed: If the uppermost card of the pile has the number $k$, then we reverse the order of the $k$ uppermost cards. Prove that, after finitely many executions of this operation, the card with the number $1$ will become the uppermost card of the pile.

1988 IMO Longlists, 48

Tags: trigonometry , geometry , perimeter

Consider 2 concentric circle radii $ R$ and $ r$ ($ R > r$) with centre $ O.$ Fix $ P$ on the small circle and consider the variable chord $ PA$ of the small circle. Points $ B$ and $ C$ lie on the large circle; $ B,P,C$ are collinear and $ BC$ is perpendicular to $ AP.$ [b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal? [b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?

2022 Assam Mathematical Olympiad, 2

Tags:

Find the sum of all the positive divisors of $27000$.

2022 Princeton University Math Competition, B1

Tags: geometry

A triangle $\vartriangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\angle EBC = 25^o$, $\angle BCA = 32^o$, and $\angle CAB = 60^o$. Find the smallest possible value of $\angle DAB$ in degrees.

2007 Nicolae Coculescu, 2

Tags: newton s binom , diophantine , equation

[b]a)[/b] Prove that there exists two infinite sequences $ \left( a_n \right)_{n\ge 1} ,\left( b_n \right)_{n\ge 1} $ of nonnegative integers such that $ a_n>b_n $ and $ (2+\sqrt 3)^n =a_n (2+\sqrt 3) -b_n , $ for any natural numbers $ n. $ [b]b)[/b] Prove that the equation $ x^2-4xy+y^2=1 $ has infinitely many solutions in $ \mathbb{N}^2. $ [i]Florian Dumitrel[/i]

1950 Poland - Second Round, 6

Tags: number theory , diophantine equation , diophantine

Solve the equation in integer numbers $$y^3-x^3=91$$

1992 National High School Mathematics League, 3

Tags: geometry , 3d geometry , tetrahedron

Areas of four surfaces of a tetrahedron are $S_1,S_2,S_3,S_4$. And the largest one of them is $S$. $\lambda=\frac{S_1+S_2+S_3+S_4}{S}$, then $\lambda$ always satisfies $\text{(A)}2<\lambda\leq4\qquad\text{(B)}3<\lambda<4\qquad\text{(C)}2.5<\lambda\leq4.5\qquad\text{(D)}3.5<\lambda<5.5$

2017 ASDAN Math Tournament, 3

Tags: algebra test

Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$. Find $ab$.

2014 Singapore Senior Math Olympiad, 9

Tags: absolute value

Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$. $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $

2017 Online Math Open Problems, 23

Tags:

Call a nonempty set $V$ of nonzero integers \emph{victorious} if there exists a polynomial $P(x)$ with integer coefficients such that $P(0)=330$ and that $P(v)=2|v|$ holds for all elements $v\in V$. Find the number of victorious sets. [i]Proposed by Yannick Yao[/i]

1983 IMO Longlists, 33

Tags: algebra , polynomial , sequence , coefficient

Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$

2008 Mathcenter Contest, 5

Tags: irrational number , algebra

There are $6$ irrational numbers. Prove that there are always three of them, suppose $a,b,c$ such that $a+b$,$b+c$,$c+a$ are irrational numbers. [i](Erken)[/i]

2019 Tuymaada Olympiad, 4

Tags: operation , algebra , combinatorics , inequalities

A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.

2015 BMT Spring, 14

Tags: algebra

Determine $$ \left|\prod^{10}_{k=1}(e^{\frac{i \pi}{2^k}}+ 1) \right|$$

2022 AMC 10, 3

Tags: algebra

The sum of three numbers is $96$. The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers? $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , inequalities

Let $a, b, c, d \ge 0$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Prove that $$\frac{a + b + c + d}{2} \ge 1 + \sqrt{abcd}$$ When does the equality hold? Leonard Giugiuc and Valmir B. Krasniqi

2008 AIME Problems, 5

Tags: trapezoid , ratio , circumcircle , analytic geometry , geometric transformation , homothety , geometry

In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$.

2021 Saint Petersburg Mathematical Olympiad, 2

Tags: number theory

Given are $2021$ prime numbers written in a row. Each number, except for those in the two ends, differs from its two adjacent numbers with $6$ and $12$. Prove that there are at least two equal numbers.

1988 IMO Longlists, 14

Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

1995 Dutch Mathematical Olympiad, 3

Tags:

Let $ 101$ marbles be numbered from $ 1$ to $ 101$. The marbles are divided over two baskets $ A$ and $ B$. The marble numbered $ 40$ is in basket $ A$. When this marble is removed from basket $ A$ and put in $ B$, the averages of the numbers $ A$ and $ B$ both increase by $ \frac{1}{4}$. How many marbles were there originally in basket $ A?$

2018 Azerbaijan BMO TST, 3

Tags: geometry , pentagon , inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2010 Contests, 3

Tags:

How many ordered triples of integers $(x, y, z)$ are there such that \[ x^2 + y^2 + z^2 = 34 \, ? \]

2015 Ukraine Team Selection Test, 9

Tags: heptagon , pentagon , convex , combinatorics , combinatorial geometry , point

The set $M$ consists of $n$ points on the plane and satisfies the conditions: $\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon, $\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon. Find the smallest possible value that $n$ can take .

1975 Miklós Schweitzer, 9

Tags: trigonometry , real analysis

Let $ l_0,c,\alpha,g$ be positive constants, and let $ x(t)$ be the solution of the differential equation \[ ([l_0\plus{}ct^{\alpha}] ^2x')'\plus{}g[l_0\plus{}ct^{\alpha}] \sin x\equal{}0, \;t \geq 0,\ \;\minus{}\frac{\pi}{2} <x< \frac{\pi}{2},\] satisfying the initial conditions $ x(t_0)\equal{}x_0, \;x'(t_0)\equal{}0$. (This is the equation of the mathematical pendulum whose length changes according to the law $ l\equal{}l_0\plus{}ct^{\alpha}$.) Prove that $ x(t)$ is defined on the interval $ [t_0,\infty)$; furthermore, if $ \alpha >2$ then for every $ x_0 \not\equal{} 0$ there exists a $ t_0$ such that \[ \liminf_{t \rightarrow \infty} |x(t)| >0.\] [i]L. Hatvani[/i]

2017 NIMO Problems, 4

Tags: number theory

For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$? [i]Proposed by Michael Tang[/i]