Olympiad problems

2016 India Regional Mathematical Olympiad, 6

Tags: number theory

Show that the infinite arithmetic progression $\{1,4,7,10 \ldots\}$ has infinitely many 3 -term sub sequences in harmonic progression such that for any two such triples $\{a_1, a_2 , a_3 \}$ and $\{b_1, b_2 ,b_3\}$ in harmonic progression , one has $$\frac{a_1} {b_1} \ne \frac {a_2}{b_2}$$.

1989 IMO Longlists, 1

Tags: algebra

In the set $ S_n \equal{} \{1, 2,\ldots ,n\}$ a new multiplication $ a*b$ is defined with the following properties: [b](i)[/b] $ c \equal{} a * b$ is in $ S_n$ for any $ a \in S_n, b \in S_n.$ [b](ii)[/b] If the ordinary product $ a \cdot b$ is less than or equal to $ n,$ then $ a*b \equal{} a \cdot b.$ [b](iii)[/b] The ordinary rules of multiplication hold for $ *,$ i.e.: [b](1)[/b] $ a * b \equal{} b * a$ (commutativity) [b](2)[/b] $ (a * b) * c \equal{} a * (b * c)$ (associativity) [b](3)[/b] If $ a * b \equal{} a * c$ then $ b \equal{} c$ (cancellation law). Find a suitable multiplication table for the new product for $ n \equal{} 11$ and $ n \equal{} 12.$

PEN G Problems, 20

Tags: irrational number , floor function , logarithm

You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.

2009 China Team Selection Test, 3

Tags: binomial coefficient , polynomial , induction , modular arithmetic , algebra

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

1999 Putnam, 4

Tags: calculus , derivative

Sum the series \[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]

2022 ISI Entrance Examination, 9

Tags: complex number , algebra

Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) .$$ for every positive integer $n \geqslant 2$ and every choice $z_{1}, \ldots, z_{n}$ of complex numbers with non-negative real and imaginary parts. [Hint: First find $k$ that works for $n=2$. Then show that the same $k$ works for any $n \geqslant 2$.]

PEN D Problems, 18

Tags: congruence , polynomial , modular arithmetic , function , algebra

Let $p$ be a prime number. Determine the maximal degree of a polynomial $T(x)$ whose coefficients belong to $\{ 0,1,\cdots,p-1 \}$, whose degree is less than $p$, and which satisfies \[T(n)=T(m) \; \pmod{p}\Longrightarrow n=m \; \pmod{p}\] for all integers $n, m$.

2017 Brazil Undergrad MO, 5

Tags: linear algebra , matrix , vector geometry , math olympiads

Let $d\leq n$ be positive integers and $A$ a real $d\times n$ matrix. Let $\sigma(A)$ be the supremum of $\inf_{v\in W,|v|=1}|Av|$ over all subspaces $W$ of $R^n$ with dimension $d$. For each $j \leq d$, let $r(j) \in \mathbb{R}^n$ be the $j$th row vector of $A$. Show that: \[\sigma(A) \leq \min_{i\leq d} d(r(i), \langle r(j), j\ne i\rangle) \leq \sqrt{n}\sigma(A)\] In which all are euclidian norms and $d(r(i), \langle r(j), j\ne i\rangle)$ denotes the distance between $r(i)$ and the span of $r(j), 1 \leq j \leq d, j\ne i$.

2023 AMC 12/AHSME, 24

Tags: number theory , least common multiple , greatest common divisor

Integers $a, b, c, d$ satisfy the following: $abcd=2^6\cdot 3^9\cdot 5^7$ $\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$ $\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$ $\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$ $\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$ $\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$ $\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$ Find $\text{gcd}(a,b,c,d)$ $\textbf{(A)}~30\qquad\textbf{(B)}~45\qquad\textbf{(C)}~3\qquad\textbf{(D)}~15\qquad\textbf{(E)}~6$

2018 HMNT, 9

Tags: geometry

Circle $\omega_1$ of radius $1$ and circle $\omega_2$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_1$ and regular pentagon $MONEY$ in $\omega_2$. It then writes down all 20 (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all together. What is the maximum possible value of his result?

2005 Tournament of Towns, 3

Tags:

There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal number of moves required for interchanging the Black and White Queens? [i](5 points)[/i]

2010 AMC 10, 17

Tags:

Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city? $ \textbf{(A)}\ 22\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 26$

1985 Tournament Of Towns, (082) T3

Tags: algebra , system of equations

Find all real solutions of the system of equations $\begin{cases} (x + y) ^3 = z \\ (y + z) ^3 = x \\ ( z+ x) ^3 = y \end{cases} $ (Based on an idea by A . Aho , J. Hop croft , J. Ullman )

2014 JHMMC 7 Contest, 16

Tags: integer , poet s strategy

The sum of two integers is $8$. The sum of the squares of those two integers is $34$. What is the product of the two integers?

1988 IMO Longlists, 88

Tags: geometry

Seven circles are given. That is, there are six circles inside a fixed circle, each tangent to the fixed circle and tangent to the two other adjacent smaller circles. If the points of contact between the six circles and the larger circle are, in order, $A_1, A_2, A_3, A_4, A_5$ and $A_6$ prove that \[ A_1 A_2 \cdot A_3 A_4 \cdot A_5 A_6 = A_2 A_3 \cdot A_4 A_5 \cdot A_6 A_1. \]

2024 Kyiv City MO Round 1, Problem 1

Tags: number theory , algebra , divisibility

Find all pairs of positive integers $(a, b)$ such that $4b - 1$ is divisible by $3a + 1$ and $3a - 1$ is divisible by $2b + 1$.

2025 Kyiv City MO Round 1, Problem 1

Tags: number theory , bruteforce , algebra

Find all three-digit numbers that are $ 5 $ times greater than the product of their digits.

1989 IMO Shortlist, 20

Tags: combinatorics , geometric inequality , combinatorial inequality , point set

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that [b]i.)[/b] no three points of $ S$ are collinear, and [b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

2011 AMC 12/AHSME, 25

Tags: geometry , incenter , circumcircle , trigonometry , function , inequalities , calculus

Triangle $ABC$ has $\angle BAC=60^\circ$, $\angle CBA \le 90^\circ$, $BC=1$, and $AC \ge AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of the pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? $\textbf{(A)}\ 60 ^\circ \qquad \textbf{(B)}\ 72 ^\circ\qquad \textbf{(C)}\ 75 ^\circ \qquad \textbf{(D)}\ 80 ^\circ\qquad \textbf{(E)}\ 90 ^\circ$

1979 Polish MO Finals, 2

Tags: geometry , 3d geometry , tetrahedron

Prove that the four lines, joining the vertices of a tetrahedron with the incenters of the opposite faces, have a common point if and only if the three products of the lengths of opposite sides are equal.

1996 Italy TST, 2

Tags: combinatorics , boolean lattice method , set

2. Let $A_1,A_2,...,A_n$be distinct subsets of an n-element set $ X$ ($n \geq 2$). Show that there exists an element $x$ of $X$ such that the sets $A_1\setminus \{x\}$ ,:......., $A_n\setminus \{x\}$ are all distinct.

2013 Stanford Mathematics Tournament, 12

Tags:

Suppose Robin and Eddy walk along a circular path with radius $r$ in the same direction. Robin makes a revolution around the circular path every $3$ minutes and Eddy makes a revolution every minute. Jack stands still at a distance $R>r$ from the center of the circular path. At time $t=0$, Robin and Eddy are at the same point on the path, and Jack, Robin, and Eddy, and the center of the path are collinear. When is the next time the three people (but not necessarily the center of the path) are collinear?

2016 Bundeswettbewerb Mathematik, 3

Tags: perpendicular , chord , geometry , circles

Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$. The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.

1987 China Team Selection Test, 1

Tags: geometry , rectangle , ratio , analytic geometry , symmetry , hyperbola , conic

Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

2017 CMIMC Individual Finals, 1

Tags: algebra

Find all real numbers $x$ such that the expression \[\log_2 |1 + \log_2 |2 + \log_2 |x| | |\] does not have a defined value.