Olympiad problems

2000 Brazil National Olympiad, 2

Let $s(n)$ be the sum of all positive divisors of $n$, so $s(6) = 12$. We say $n$ is almost perfect if $s(n) = 2n - 1$. Let $\mod(n, k)$ denote the residue of $n$ modulo $k$ (in other words, the remainder of dividing $n$ by $k$). Put $t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)$. Show that $n$ is almost perfect if and only if $t(n) = t(n-1)$.

2008 Mathcenter Contest, 4

Tags: inequalities , algebra

Let $p,q,r \in \mathbb{R}^+$ and for every $n \in \mathbb{N}$ where $pqr=1$ , denote $$ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+ 1} \leq 1$$ [i](Art-Ninja)[/i]

2016 Purple Comet Problems, 5

Tags:

Julius has a set of ﬁve positive integers whose mean is 100. If Julius removes the median of the set of ﬁve numbers, the mean of the set increases by 5, and the median of the set decreases by 5. Find the maximum possible value of the largest of the ﬁve numbers Julius has.

2023 Tuymaada Olympiad, 4

Tags: combinatorics , game theory , combinatorial game theory

Two players play a game. They have $n > 2$ piles containing $n^{10}+1$ stones each. A move consists of removing all the piles but one and dividing the remaining pile into $n$ nonempty piles. The player that cannot move loses. Who has a winning strategy, the player that moves first or his adversary?

2018 IMAR Test, 1

Tags: locus , geometry

Let $ABC$ be a triangle whose angle at $A$ is right, and let $D$ be the foot of the altitude from $A$. A variable point $M$ traces the interior of the minor arc $AB$ of the circle $ABC$. The internal bisector of the angle $DAM$ crosses $CM$ at $N$. The line through $N$ and perpendicular to $CM$ crosses the line $AD$ at $P$. Determine the locus of the point where the line $BN$ crosses the line $CP$. [i]* * *[/i]

2018 Thailand TSTST, 2

Tags: combinatorics

There are three sticks, each of which has an integer length which is at least $n$; the sum of their lengths is $n(n + 1)/2$. Prove that it is possible to break the sticks (possibly several times) so that the resulting sticks have length $1, 2,\dots, n$. [i]Note: a stick of length $a + b$ can be broken into sticks of lengths $a$ and $b$.[/i]

1991 Tournament Of Towns, (286) 2

Tags: geometry , pentagon , perpendicular

The pentagon $ABCDE$ has an inscribed circle and the diagonals $AD$ and $CE$ intersect in its centre $O$. Prove that the segment $BO$ and the side $DE$ are perpendicular. (Folklore)

2010 Harvard-MIT Mathematics Tournament, 7

Tags: geometry

You are standing in an infinitely long hallway with sides given by the lines $x=0$ and $x=6$. You start at $(3,0)$ and want to get to $(3,6)$. Furthermore, at each instant you want your distance to $(3,6)$ to either decrease or stay the same. What is the area of the set of points that you could pass through on your journey from $(3,0)$ to $(3,6)$?

2011 Canadian Open Math Challenge, 6

Tags:

Integers a, b, c, d, and e satisfy the following three properties: (i) $2 \le a < b <c <d <e <100$ (ii)$ \gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence. What is the value of c?

2021 JHMT HS, 5

Tags: general

Terry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. However, he accidentally swaps the digits of one of the numbers, and thus gets the incorrect sum of $4941.$ What is the largest possible number whose digits Terry could have swapped in the summation?

2011 Indonesia TST, 4

Tags: number theory

Given an arbitrary prime $p>2011$. Prove that there exist positive integers $a, b, c$ not all divisible by $p$ such that for all positive integers $n$ that $p\mid n^4- 2n^2+ 9$, we have $p\mid 24an^2 + 5bn + 2011c$.

2025 Bangladesh Mathematical Olympiad, P2

Tags: equation , algebra , exponential equation

Find all real solutions to the equation $(x^2-9x+19)^{x^2-3x+2} = 1$.

2015 Federal Competition For Advanced Students, 1

Tags: algebra , inequalities

Let $a$, $b$, $c$, $d$ be positive numbers. Prove that $$(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)$$ When does equality hold? (Georg Anegg)

2024 Harvard-MIT Mathematics Tournament, 17

Tags: guts

The numbers $1, 2, \ldots, 20$ are put into a hat. Claire draws two numbers from the hat uniformly at random, $a<b,$ and then puts them back into the hat. Then, William draws two numbers from the hat uniformly at random, $c<d.$ Let $N$ denote the number of integers $n$ that satisfy exactly one of $a \le n \le b$ and $c \le n \le d.$ Compute the probability that $N$ is even.

2014 Contests, 2

Tags: circumcircle , incenter , cyclic quadrilateral , geometry

Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$. Prove \[ AN \cdot NC = CD \cdot BN. \]

2017 CMIMC Number Theory, 9

Tags: number theory

Find the smallest prime $p$ for which there exist positive integers $a,b$ such that \[ a^{2} + p^{3} = b^{4}. \]

2022 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics , graph theory , algebra

Given is a graph $G$ of $n+1$ vertices, which is constructed as follows: initially there is only one vertex $v$, and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that $v$ has weight $0$ and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by $1$. Prove that no weight can exceed $n^2$.

2022 Silk Road, 3

Tags: algebra

In an infinite sequence $\{\alpha\}, \{\alpha^2\}, \{\alpha^3\}, \cdots $ there are finitely many distinct values$.$ Show that $\alpha$ is an integer$. (\{x\}$ denotes the fractional part of$ x.)$ [i](Golovanov A.S.)[/i]

2021 European Mathematical Cup, 4

Tags: polynomial , algebra

Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and $$P(x)^2+1=(x^2+1)Q(x)^2.$$

2021 Girls in Math at Yale, R6

Tags: college

16. Suppose trapezoid $JANE$ is inscribed in a circle of radius $25$ such that the center of the circle lies inside the trapezoid. If the two bases of $JANE$ have side lengths $14$ and $30$ and the average of the lengths of the two legs is $\sqrt{m}$, what is $m$? 17. What is the radius of the circle tangent to the $x$-axis, the line $y=\sqrt{3}x$, and the circle $(x-10\sqrt{3})^2+(y-10)^2=25$? 18. Find the smallest positive integer $n$ such that $3n^3-9n^2+5n-15$ is divisible by $121$ but not $2$.

Estonia Open Senior - geometry, 2008.2.3

Tags: geometry , parallelogram , circles

Two circles are drawn inside a parallelogram $ABCD$ so that one circle is tangent to sides $AB$ and $AD$ and the other is tangent to sides $CB$ and $CD$. The circles touch each other externally at point $K$. Prove that $K$ lies on the diagonal $AC$.

2008 Irish Math Olympiad, 2

Tags: trapezoid , geometry

Circles $ S$ and $ T$ intersect at $ P$ and $ Q$, with $ S$ passing through the centre of $ T$. Distinct points $ A$ and $ B$ lie on $ S$, inside $ T$, and are equidistant from the centre of $ T$. The line $ PA$ meets $ T$ again at $ D$. Prove that $ |AD| \equal{} |PB|$.

2021 BMT, 13

Tags: combinatorics

How many ways are there to completely fill a $3 \times 3$ grid of unit squares with the letters $B, M$, and $T$, assigning exactly one of the three letters to each of the squares, such that no $2$ adjacent unit squares contain the same letter? Two unit squares are adjacent if they share a side.

1990 ITAMO, 5

Tags: number theory , divisible , trinomial

Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.

2015 Iberoamerican Math Olympiad, 3

Tags: algebra , polynomial

Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$: $s_n = \alpha^n + \beta^n$ $t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$ Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.