Olympiad problems

2002 Putnam, 6

Tags: linear algebra , matrix , algebra , polynomial , modular arithmetic , binomial theorem

Let $p$ be a prime number. Prove that the determinant of the matrix \[ \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} \] is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a$, $b$, and $c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)

2004 VJIMC, Problem 3

Tags: geometry , real analysis , sequence

Denote by $B(c,r)$ the open disk of center $c$ and radius $r$ in the plane. Decide whether there exists a sequence $\{z_n\}^\infty_{n=1}$ of points in $\mathbb R^2$ such that the open disks $B(z_n,1/n)$ are pairwise disjoint and the sequence $\{z_n\}^\infty_{n=1}$ is convergent.

2008 Moldova Team Selection Test, 2

Tags: number theory

Let $ p$ be a prime number and $ k,n$ positive integers so that $ \gcd(p,n)\equal{}1$. Prove that $ \binom{n\cdot p^k}{p^k}$ and $ p$ are coprime.

2016 Romania National Olympiad, 2

Tags: linear algebra , rank

Consider a natural number, $ n\ge 2, $ and three $ n\times n $ complex matrices $ A,B,C $ such that $ A $ is invertible, $ B $ is formed by replacing the first line of $ A $ with zeroes, and $ C $ is formed by putting the last $ n-1 $ lines of $ A $ above a line of zeroes. Prove that: [b]a)[/b] $ \text{rank} \left( A^{-1} B \right) = \text{rank} \left( \left( A^{-1} B\right)^2 \right) =\cdots =\text{rank} \left( \left( A^{-1} B\right)^n \right) $ [b]b)[/b] $ \text{rank} \left( A^{-1} C \right) > \text{rank} \left( \left( A^{-1} C\right)^2 \right) >\cdots >\text{rank} \left( \left( A^{-1} C\right)^n \right) $

2022 China Team Selection Test, 1

Tags: geometry , circles , equal angles

Given two circles $\omega_1$ and $\omega_2$ where $\omega_2$ is inside $\omega_1$. Show that there exists a point $P$ such that for any line $\ell$ not passing through $P$, if $\ell$ intersects circle $\omega_1$ at $A,B$ and $\ell$ intersects circle $\omega_2$ at $C,D$, where $A,C,D,B$ lie on $\ell$ in this order, then $\angle APC=\angle BPD$.

2006 JBMO ShortLists, 4

Tags: parameterization , number theory

Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.

2010 Tournament Of Towns, 5

Tags: combinatorics

In a tournament with $55$ participants, one match is played at a time, with the loser dropping out. In each match, the numbers of wins so far of the two participants differ by not more than $1$. What is the maximal number of matches for the winner of the tournament?

1998 Harvard-MIT Mathematics Tournament, 1

Tags: trigonometry

Evaluate $\sin(1998^\circ+237^\circ)\sin(1998^\circ-1653^\circ)$.

2010 Mathcenter Contest, 4

Tags: geometry , construction

In a circle, two non-intersecting chords $AB,CD$ are drawn.On the chord $AB$,a point $E$ (different from $A$,$B$) is taken Consider the arc $AB$ that does not contain the points $C,D$. With a compass and a straighthedge, find all possible point $F$ on that arc such that $\dfrac{PE}{EQ}=\dfrac{1}{2}$, where $P$ and $Q$ are the points in which the chord $AB$ meets the segment $FC$ and $FD$. [i](tatari/nightmare)[/i]

2008 Middle European Mathematical Olympiad, 2

Tags: algorithm , function , combinatorics

On a blackboard there are $ n \geq 2, n \in \mathbb{Z}^{\plus{}}$ numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers $ n$ for which it is possible to yield $ n$ identical number after a finite number of steps.

2005 Finnish National High School Mathematics Competition, 4

Tags: modular arithmetic , number theory

The numbers $1, 3, 7$ and $9$ occur in the decimal representation of an integer. Show that permuting the order of digits one can obtain an integer divisible by $7.$

1989 IMO, 5

Tags: modular arithmetic , number theory , prime number , sequence , power of number

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

2017 239 Open Mathematical Olympiad, 1

Tags: permutation , combinatorics

Denote every permutation of $1,2,\dots, n$ as $\sigma =(a_1,a_2,\dots,n)$. Prove that the sum $$\sum \frac{1}{(a_1)(a_1+a_2)(a_1+a_2+a_3)\dots(a_1+a_2+\dots+a_n)}$$ taken over all possible permutations $\sigma$ equals $\frac{1}{n!}$.

Estonia Open Senior - geometry, 2013.2.3

Tags: geometry , concyclic , circles

Circles $c_1, c_2$ with centers $O_1, O_2$, respectively, intersect at points $P$ and $Q$ and touch circle c internally at points $A_1$ and $A_2$, respectively. Line $PQ$ intersects circle c at points $B$ and $D$. Lines $A_1B$ and $A_1D$ intersect circle $c_1$ the second time at points $E_1$ and $F_1$, respectively, and lines $A_2B$ and $A_2D$ intersect circle $c_2$ the second time at points $ E_2$ and $F_2$, respectively. Prove that $E_1, E_2, F_1, F_2$ lie on a circle whose center coincides with the midpoint of line segment $O_1O_2$.

Russian TST 2018, P1

Tags: string , combinatorics , sorting , distance , extremal combinatorics , radius

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

2016 Czech And Slovak Olympiad III A, 1

Tags: number theory , sum , fraction , integer

Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.

MIPT student olimpiad spring 2024, 1

Tags: calculus , integration

Find integral: $\int_{x^2+y^2\leq 1}e^xcos(y)dxdy$

1958 Poland - Second Round, 1

Tags: prime , number theory , prime number

Prove that if $ a $ is an integer different from $ 1 $ and $ - 1 $, then $ a^4 + 4 $ is not a prime number.

Kyiv City MO Seniors 2003+ geometry, 2007.11.5

Tags: circumcircle , equal angles , fixed point , fixed , geometry

The points $A$ and $P$ are marked on the plane. Consider all such points $B, C $ of this plane that $\angle ABP = \angle MAB$ and $\angle ACP = \angle MAC $, where $M$ is the midpoint of the segment $BC$. Prove that all the circumscribed circles around the triangle $ABC$ for different points $B$ and $C$ pass through some fixed point other than the point $A$. (Alexei Klurman)

2017 Denmark MO - Mohr Contest, 3

Tags: geometry , arc , area

The figure shows an arc $\ell$ on the unit circle and two regions $A$ and $B$. Prove that the area of $A$ plus the area of $B$ equals the length of $\ell$. [img]https://1.bp.blogspot.com/-SYoSrFowZ30/XzRz0ygiOVI/AAAAAAAAMUs/0FCduUoxKGwq0gSR-b3dtb3SvDjZ89x_ACLcBGAsYHQ/s0/2017%2BMohr%2Bp3.png[/img]

2023 Dutch BxMO TST, 3

Tags: combinatorics

We play a game of musical chairs with $n$ chairs numbered $1$ to $n$. You attach $n$ leaves, numbered $1$ to $n$, to the chairs in such a way that the number on a leaf does not match the number on the chair it is attached to. One player sits on each chair. Every time you clap, each player looks at the number on the leaf attached to his current seat and moves to sit on the seat with that number. Prove that, for any $m$ that is not a prime power with$ 1 < m \leq n$, it is possible to attach the leaves to the seats in such a way that after $m$ claps everyone has returned to the chair they started on for the first time.

2000 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra , inequalities

Find all real numbers $ a $ such that $ x,y>a\implies x+y+xy>a. $ [i]Gheorghe Iurea[/i]

1998 Chile National Olympiad, 4

Tags: algebra , inequalities

a) Prove that for any nonnegative real $x$, holds $$x^{\frac32} + 6x^{\frac54} + 8x^{\frac34}\ge 15x.$$ b) Determine all x for which the equality holds

2023 Iran MO (3rd Round), 1

Tags: geometry , incenter , circumcircle , angle bisector

In triangle $\triangle ABC$ , $I$ is the incenter and $M$ is the midpoint of arc $(BC)$ in the circumcircle of $(ABC)$not containing $A$. Let $X$ be an arbitrary point on the external angle bisector of $A$. Let $BX \cap (BIC) = T$. $Y$ lies on $(AXC)$ , different from $A$ , st $MA=MY$ . Prove that $TC || AY$ (Assume that $X$ is not on $(ABC)$ or $BC$)

2016 Austria Beginners' Competition, 3

Tags: number theory , combinatorics

We consider the following ﬁgure: [See attachment] We are looking for labellings of the nine ﬁelds with the numbers 1, 2, ..., 9. Each of these numbers has to be used exactly once. Moreover, the six sums of three resp. four numbers along the drawn lines have to be be equal. Give one such labelling. Show that all such labellings have the same number in the top ﬁeld. How many such labellings do there exist? (Two labellings are considered diﬀerent, if they disagree in at least one ﬁeld.) (Walther Janous)