Olympiad problems

1992 Nordic, 3

Prove that among all triangles with inradius $1$, the equilateral one has the smallest perimeter .

1993 Tournament Of Towns, (365) 4

Tags: combinatorics

There are $25$ students in Peter’s class (not counting him). Peter has observed that all $25$ have different numbers of friends in this class. How many friends does Peter have in this class? (Give all possible answers.) (S Toparev)

1950 Miklós Schweitzer, 1

Tags: real analysis

Let $ a>0$, $ d>0$ and put $ f(x)\equal{}\frac{1}{a}\plus{}\frac{x}{a(a\plus{}d)}\plus{}\cdots\plus{}\frac{x^n}{a(a\plus{}d)\cdots(a\plus{}nd)}\plus{}\cdots$ Give a closed form for $ f(x)$.

1969 IMO Shortlist, 25

Tags: number theory , divisibility , frobenius , additive number theory

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

1966 IMO Shortlist, 10

Tags: trigonometric equation , trigonometry , algebra , equation

How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?

2025 All-Russian Olympiad, 11.1

Tags: complex number , algebra

$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers $a$ and $b$ from the board such that: \[ a^2 + b^2 + 1 = 2ab \] Ways that differ by the order of selection are considered the same. Prove that there exist two numbers $c$ and $d$ from the board such that: \[ c^2 + d^2 + 2025 = 2cd \]

2023 LMT Fall, 6

Tags: combinatorics

Jeff rolls a standard $6$ sided die repeatedly until he rolls either all of the prime numbers possible at least once, or all the of even numbers possible at least once. Find the probability that his last roll is a $2$.

2001 Mongolian Mathematical Olympiad, Problem 1

Tags: sequence , recurrence relation , algebra

Suppose that a sequence $x_1,x_2,\ldots,x_{2001}$ of positive real numbers satisfies $$3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.$$Find the maximum possible value of $x_1$.

2018 China Second Round Olympiad, 3

Tags: combinatorics

Let $n,k,m$ be positive integers, where $k\ge 2$ and $n\le m < \frac{2k-1}{k}n$. Let $A$ be a subset of $\{1,2,\ldots ,m\}$ with $n$ elements. Prove that every integer in the range $\left(0,\frac{n}{k-1}\right)$ can be expressed as $a-b$, where $a,b\in A$.

2006 China Northern MO, 1

Tags: projective geometry , geometry

$AB$ is the diameter of circle $O$, $CD$ is a non-diameter chord that is perpendicular to $AB$. Let $E$ be the midpoint of $OC$, connect $AE$ and extend it to meet the circle at point $P$. Let $DP$ and $BC$ meet at $F$. Prove that $F$ is the midpoint of $BC$.

2008 Iran Team Selection Test, 8

Tags: algebra , polynomial , function , limit , search , number theory , relatively prime

Find all polynomials $ p$ of one variable with integer coefficients such that if $ a$ and $ b$ are natural numbers such that $ a \plus{} b$ is a perfect square, then $ p\left(a\right) \plus{} p\left(b\right)$ is also a perfect square.

2019 China Team Selection Test, 2

Tags: graph theory , combinatorics

A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ .

Estonia Open Senior - geometry, 2018.2.5

Tags: reflection , equal angles , angle , geometry

Let $A'$ be the result of reflection of vertex $A$ of triangle ABC through line $BC$ and let $B'$ be the result of reflection of vertex $B$ through line $AC$. Given that $\angle BA' C = \angle BB'C$, can the largest angle of triangle $ABC$ be located: a) At vertex $A$, b) At vertex $B$, c) At vertex $C$?

2007 Baltic Way, 8

Tags: combinatorics

Call a set $A$ of integers [i]non-isolated[/i], if for every $a\in A$ at least one of the numbers $a-1$ and $a+1$ also belongs to $A$. Prove that the number of five-element non-isolated subsets of $\{1, 2,\ldots ,n\}$ is $(n-4)^2$.

2010 Belarus Team Selection Test, 4.1

Tags: subset , algebra

Find all finite sets $M \subset R, |M| \ge 2$, satisfying the following condition: [i]for all $a, b \in M, a \ne b$, the number $a^3 - \frac{4}{9}b$ also belongs to $M$. [/i] (I. Voronovich)

2015 Canadian Mathematical Olympiad Qualification, 3

Tags: permutation , number theory , average

Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$. Determine the largest mediocre number.

1956 Putnam, B1

Tags: differential equation

Show that if the differential equation $$M(x,y)\, dx +N(x,y) \, dy =0$$ is both homogeneous and exact, then the solution $y=y(x)$ satisfies that $xM(x,y)+yN(x,y)$ is constant.

2017 Philippine MO, 3

Tags: combinatorics , arithmetic sequence , coloring

Each of the numbers in the set $A = \{1,2, \cdots, 2017\}$ is colored either red or white. Prove that for $n \geq 18$, there exists a coloring of the numbers in $A$ such that any of its n-term arithmetic sequences contains both colors.

2016 Auckland Mathematical Olympiad, 2

Tags: geometry , equal angles , square

In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$. Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$. If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$.

2022 AMC 8 -, 13

Tags:

How many positive integers can fill the blank in the sentence below? "One positive integer is $\underline{~~~~~}$ more than twice another, and the sum of the two numbers is 28." $\textbf{(A)} ~6\qquad\textbf{(B)} ~7\qquad\textbf{(C)} ~8\qquad\textbf{(D)} ~9\qquad\textbf{(E)} ~10\qquad$

2007 Mongolian Mathematical Olympiad, Problem 1

Tags: geometry

Let $M$ be the midpoint of the side $BC$ of triangle $ABC$. The bisector of the exterior angle of point $A$ intersects the side $BC$ in $D$. Let the circumcircle of triangle $ADM$ intersect the lines $AB$ and $AC$ in $E$ and $F$ respectively. If the midpoint of $EF$ is $N$, prove that $MN\parallel AD$.

2019 Junior Balkan MO, 2

Tags: algebra , inequalities

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

2019 Tournament Of Towns, 4

Tags: number theory , consecutive , multiple , sum of squares , circles

There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)

1976 IMO, 3

Tags: geometry , 3d geometry , combinatorics , packing

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

1997 Estonia National Olympiad, 2

Tags: triangle , geometry , angle

Side lengths $a,b,c$ of a triangle satisfy $\frac{a^3+b^3+c^3}{a+b+c}= c^2$. Find the measure of the angle opposite to side $c$.