Olympiad problems

1976 Miklós Schweitzer, 10

Tags: topology , function

Suppose that $ \tau$ is a metrizable topology on a set $ X$ of cardinality less than or equal to continuum. Prove that there exists a separable and metrizable topology on $ X$ that is coarser that $ \tau$. [i]L. Juhasz[/i]

2002 Federal Competition For Advanced Students, Part 1, 4

Tags: geometry , parallelogram , geometry unsolved

Let $A,C, P$ be three distinct points in the plane. Construct all parallelograms $ABCD$ such that point $P$ lies on the bisector of angle $DAB$ and $\angle APD = 90^\circ$.

2002 China Team Selection Test, 3

Tags: number theory unsolved , number theory

For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that \[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\ b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\ c^2 &= \alpha\beta\gamma. \end{cases} \] Also, let $ \lambda$ be a real number that satisfies the condition \[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\] Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.

2003 AMC 8, 5

Tags: percent , LaTeX

If $20\%$ of a number is $12$, what is $30\%$ of the same number? $\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 30$

2009 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , geometry

Let $\mathcal{R}$ be the region in the plane bounded by the graphs of $y=x$ and $y=x^2$. Compute the volume of the region formed by revolving $\mathcal{R}$ around the line $y=x$.

1972 Dutch Mathematical Olympiad, 5

Tags: geometry , ratio

Given is an acute-angled triangle $ABC$ with angles $\alpha$, $\beta$ and $\gamma$. On side $AB$ lies a point $P$ such that the line connecting the feet of the perpendiculars from $P$ on $AC$ and $BC$ is parallel to $AB$. Express the ratio $\frac{AP}{BP}$ in terms of $\alpha$ and $\beta$.

2010 Saudi Arabia Pre-TST, 3.4

Tags: parameterization , algebra

Let $a$ and $b$ be real numbers such that $a + b \ne 0$. Solve the equation $$\frac{1}{(x + a)^2 - b^2} +\frac{1}{(x +b)^2 - a^2}=\frac{1}{x^2 -(a + b)^2}+\frac{1}{x^2-(a -b)^2}$$

2021 MMATHS, 2

Tags: Yale , MMATHS

In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's [i]score[/i] is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\times n$ grid with probability $k$; he notices that the expected value of the score of the resulting grid is equal to $k$, too! Given that $k > 0.9999$, find the minimum possible value of $n$. [i]Proposed by Andrew Wu[/i]

2022 Switzerland Team Selection Test, 1

Tags: number theory , Divisibility , number bases , Switzerland TST

Let $n$ be a positive integer. Prove that there exists a finite sequence $S$ consisting of only zeros and ones, satisfying the following property: for any positive integer $d \geq 2$, when $S$ is interpreted in base $d$, the resulting number is non-zero and divisible by $n$. [i]Remark: The sequence $S=s_ks_{k-1} \cdots s_1s_0$ interpreted in base $d$ is the number $\sum_{i=0}^{k}s_id^i$[/i]

2002 Bundeswettbewerb Mathematik, 3

Tags: geometry , induction , combinatorics proposed , combinatorics

Given a convex polyhedron with an even number of edges. Prove that we can attach an arrow to each edge, such that for every vertex of the polyhedron, the number of the arrows ending in this vertex is even.

2018 PUMaC Team Round, 5

Tags: PuMAC , Team Round

There exist real numbers $a$, $b$, $c$, $d$, and $e$ such that for all positive integers $n$, we have $$\sqrt{n}=\sum_{i=0}^{n-1}\sqrt[5]{\sqrt{ai^5+bi^4+ci^3+di^2+ei+1}-\sqrt{ai^5+bi^4+ci^3+di^2+ei}}.$$ Find $a+b+c+d$.

2022 Bulgarian Autumn Math Competition, Problem 8.2

Tags: geometry

It's given a right-angled triangle $ABC (\angle{C}=90^{\circ})$ and area $S$. Let $S_1$ be the area of the circle with diameter $AB$ and $k=\frac{S_1}{S}$\\ a) Compute the angles of $ABC$, if $k=2\pi$ b) Prove it is not possible for k to be $3$

1986 India National Olympiad, 5

Tags: algebra , polynomial , inequalities , AMC , USA(J)MO , USAMO , algebra proposed

If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.

2014 ITAMO, 5

Tags: algebra proposed , algebra

Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$-th powers of $2015$ distinct positive integers $x_1 <x_2 <\cdots <x_{2015}$.

2018 CMIMC Combinatorics, 3

Tags: 2018 , combinatorics

Michelle is at the bottom-left corner of a $6\times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$; the first time Michelle moves to one of these points she is instantly teleported to the other point and the devices disappear. If she can only move up or to the right in unit increments, in how many ways can she reach the point $(5,5)$?

1992 IMO Longlists, 76

Tags: geometry , 3D geometry , pyramid , IMO Shortlist , IMO Longlist

Given any triangle $ABC$ and any positive integer $n$, we say that $n$ is a [i]decomposable[/i] number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$. Determine the positive integers that are decomposable numbers for every triangle.

2002 National Olympiad First Round, 6

Tags: modular arithmetic , least common multiple

The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 2 \qquad\textbf{c)}\ 4 \qquad\textbf{d)}\ 6 \qquad\textbf{e)}\ 8 $

1957 Moscow Mathematical Olympiad, 371

Tags: quadrilateral construction , rectangle , construction , geometry

Given quadrilateral $ABCD$ and the directions of its sides. Inscribe a rectangle in $ABCD$.

2006 Peru IMO TST, 2

Tags: LaTeX , function , algebra proposed , algebra

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 02[/b] Find all pairs $(a,b)$ real positive numbers $a$ and $b$ such that: $[a[bn]]= n-1,$ for all $n$ positive integer. Note: [x] denotes the integer part of $x$. ---------- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88510]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

2013 Tournament of Towns, 3

Tags: geometry , Equilateral , circumcircle , Concyclic

Let $ABC$ be an equilateral triangle with centre $O$. A line through $C$ meets the circumcircle of triangle $AOB$ at points $D$ and $E$. Prove that points $A, O$ and the midpoints of segments $BD, BE$ are concyclic.

2005 Tournament of Towns, 1

Tags: analytic geometry

On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis. [i](4 points)[/i]

1996 Tournament Of Towns, (502) 5

Tags: number theory

Prove that there exist an infinite number of triples $n-1 $,$n$,$n + 1$ such that (a) $n$ can be represented as the sum of two squares of natural numbers but neither of $n-1$ and $n+1$ can; (b) each of these three numbers can be represented as the sum of two squares. (V Senderov)

2016 Korea USCM, 3

Tags: linear algebra , matrix , trace , invertible , college contests

Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries. (1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.) (2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.

V Soros Olympiad 1998 - 99 (Russia), 11.2

Tags: algebra , inequalities

Find the greatest value of $C$ for which, for any $x, y, z,u$, and such that for $0\le x\le y \le z\le u$, holds the inequality $$(x + y +z + u)^2 \ge Cyz .$$

2008 Middle European Mathematical Olympiad, 1

Tags: inequalities , algebra unsolved , algebra

Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the least possible value of $ a_{2008}.$