Olympiad problems

1985 IMO Longlists, 18

The circles $(R, r)$ and $(P, \rho)$, where $r > \rho$, touch externally at $A$. Their direct common tangent touches $(R, r)$ at B and $(P, \rho)$ at $C$. The line $RP$ meets the circle $(P, \rho)$ again at $D$ and the line $BC$ at $E$. If $|BC| = 6|DE|$, prove that: [b](a)[/b] the lengths of the sides of the triangle $RBE$ are in an arithmetic progression, and [b](b)[/b] $|AB| = 2|AC|.$

1953 Putnam, B1

Tags: series , convergence

Is the infinite series $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$ convergent?

1970 AMC 12/AHSME, 3

Tags:

If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is: $\textbf{(A) }\dfrac{x+1}{x-1}\qquad\textbf{(B) }\dfrac{x+2}{x-1}\qquad\textbf{(C) }\dfrac{x}{x-1}\qquad\textbf{(D) }2-x\qquad \textbf{(E) }\dfrac{x-1}{x}$

2019 Putnam, A2

Tags:

In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2\tan^{-1}(1/3)$. Find $\alpha$.

2012 Israel National Olympiad, 5

Tags: number theory , diophantine equation

Find all integer solutions of the equation $a^3+3ab^2+7b^3=2011$.

2011 Germany Team Selection Test, 3

Tags: algebra , function , functional equation

We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$ a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$ b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$

2024 Dutch IMO TST, 1

Tags: coprime , number theory , divisor

For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

2005 Germany Team Selection Test, 3

Tags: inequalities , modular arithmetic , algorithm , induction , number theory

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

2011 Purple Comet Problems, 8

Tags: geometry

A square measuring $15$ by $15$ is partitioned into five rows of five congruent squares as shown below. The small squares are alternately colored black and white as shown. Find the total area of the part colored black. [asy] size(150); defaultpen(linewidth(0.8)); int i,j; for(i=1;i<=5;i=i+1) { for(j=1;j<=5;j=j+1) { if (floor((i+j)/2)==((i+j)/2)) { filldraw(shift((i-1,j-1))*unitsquare,gray); } else { draw(shift((i-1,j-1))*unitsquare); } } } [/asy]

2021 Science ON all problems, 1

Tags: number theory

Consider the prime numbers $p_1,p_2,\dots ,p_{2021}$ such that the sum $$p_1^4+p_2^4+\dots +p_{2021}^4$$ is divisible by $6060$. Prove that at least $4$ of these prime numbers are less than $2021$. $\textit{Stefan Bălăucă}$

2019 ELMO Shortlist, N4

Tags: number theory , divisibility

A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$. [i]Proposed by Carl Schildkraut and Holden Mui[/i]

2007 Pre-Preparation Course Examination, 9

Tags: modular arithmetic , number theory

Solve the equation $4xy-x-y=z^2$ in positive integers.

2013 ELMO Shortlist, 7

Tags: ratio , geometry , parallelogram , circumcircle , trigonometry , collinear , complex

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

2021 BMT, 11

Tags: number theory

Compute the sum of all prime numbers $p$ with $p \ge 5$ such that $p$ divides $(p + 3)^{p-3} + (p + 5)^{p-5}$. .

1998 Brazil Team Selection Test, Problem 3

Tags: combinatorics , number theory , algebra

Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.

2013 Cono Sur Olympiad, 4

Tags: induction , combinatorics

Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is given one of $k$ available colors, with the only condition that if the union of two different subsets $A$ and $B$ is $M$, then $A$ and $B$ are given different colors. What is the least possible value of $k$?

2010 Ukraine Team Selection Test, 4

Tags: inequalities , algebra

For the nonnegative numbers $a, b, c$ prove the inequality: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52$$

2012 India IMO Training Camp, 2

Tags: pigeonhole principle , number theory

Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.

2005 Sharygin Geometry Olympiad, 10

Tags: geometry , similar triangles , cut

Cut the non-equilateral triangle into four similar triangles, among which not all are the same.

2009 Indonesia TST, 4

Tags: incenter , circumcircle , geometry

Given triangle $ ABC$ with $ AB>AC$. $ l$ is tangent line of the circumcircle of triangle $ ABC$ at $ A$. A circle with center $ A$ and radius $ AC$, intersect $ AB$ at $ D$ and $ l$ at $ E$ and $ F$. Prove that the lines $ DE$ and $ DF$ pass through the incenter and excenter of triangle $ ABC$.

2016 AIME Problems, 6

Tags: geometry , fact 5 , power of a point , radical axis

In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $AB$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI = 2$ and $LD = 3$, then $IC = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

2001 Romania National Olympiad, 1

Tags: function , calculus , derivative , real analysis

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, derivable on $R\backslash\{x_0\}$, having finite side derivatives in $x_0$. Show that there exists a derivable function $g:\mathbb{R}\rightarrow\mathbb{R}$, a linear function $h:\mathbb{R}\rightarrow\mathbb{R}$ and $\alpha\in\{-1,0,1\}$ such that: \[ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} \]

2016 CMIMC, 7

Tags: team

In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.

2007 China Northern MO, 4

Tags: combinatorics

For every point on the plane, one of $ n$ colors are colored to it such that: $ (1)$ Every color is used infinitely many times. $ (2)$ There exists one line such that all points on this lines are colored exactly by one of two colors. Find the least value of $ n$ such that there exist four concyclic points with pairwise distinct colors.

1983 IMO Shortlist, 19

Tags: algebra , polynomial , fibonacci , fibonacci sequence , sequence

Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying \[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\] Prove that $P(1983) = F_{1983} - 1.$