Olympiad problems

2024 Sharygin Geometry Olympiad, 19

A triangle $ABC$, its circumcircle, and its incenter $I$ are drawn on the plane. Construct the circumcenter of $ABC$ using only a ruler.

2014 India National Olympiad, 4

Tags: quadratics , algebra , polynomial , calculus , integration , probability , analytic geometry

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

2019 BMT Spring, Tie 2

Tags: number theory

Find the sum of first two integers $n > 1$ such that $3^n$ is divisible by $n$ and $3^n - 1$ is divisible by $n - 1$.

1996 Bulgaria National Olympiad, 2

Tags: geometry

Find the side length of the smallest equilateral triangle in which three discs of radii $2,3,4$ can be placed without overlap.

OMMC POTM, 2022 4

Tags: number theory , ommc , prime factorization

Define a function $P(n)$ from the set of positive integers to itself, where $P(1)=1$ and if an integer $n > 1$ has prime factorization $$n = p_1^{a_1}p_2^{a_2} \dots p_k^{a_k}$$ then $$P(n) = a_1^{p_1}a_2^{p_2} \dots a_k^{p_k}.$$ Prove that $P(P(n)) \le n$ for all positive integers $n.$ [i]Proposed by Evan Chang (squareman), USA[/i]

2015 Peru Cono Sur TST, P1

Tags: combinatorics

$A$ writes, at his choice, $8$ ones and $8$ twos on a $4\times 4$ board. Then $B$ covers the board with $8$ dominoes and for each domino she finds the smaller of the two numbers that that domino covers. Finally, $A$ adds these $8$ numbers and the result is her score. What is the highest score $A$ can secure, no matter how $B$ plays? Clarification: A domino is a $1\times 2$ or $2\times 1$ rectangle that covers exactly two squares on the board.

1991 Irish Math Olympiad, 2

Tags: inequalities

Let $$a_n=\frac{n^2+1}{\sqrt{n^4+4}}, \quad n=1,2,3,\dots$$ and let $b_n$ be the product of $a_1,a_2,a_3,\dots ,a_n$. Prove that $$\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},$$ and deduce that $$\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}$$ for all positive integers $n$.

2016 APMC, 6

Tags: number theory

Let $a$ be a natural number, $a>3$. Prove there is an infinity of numbers n, for which $a+n|a^{n}+1$

1990 Bulgaria National Olympiad, Problem 1

Tags: combinatorics

Consider the number obtained by writing the numbers $1,2,\ldots,1990$ one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?

2014 Junior Regional Olympiad - FBH, 2

Tags: Fractions

In one class in the school, number of abscent students is $\frac{1}{6}$ of number of students who were present. When teacher sent one student to bring chalk, number of abscent students was $\frac{1}{5}$ of number of students who were present. How many students are in that class?

II Soros Olympiad 1995 - 96 (Russia), 11.3

Tags: algebra , system of equations , trigonometry

Solve the system of equations $$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$

2014 Singapore Senior Math Olympiad, 10

Tags: function

If $f(x)=\frac{1}{x}-\frac{4}{\sqrt{x}}+3$ where $\frac{1}{16}\le x\le 1$, find the range of $f(x)$. $ \textbf{(A) }-2\le f(x)\le 4 \qquad\textbf{(B) }-1\le f(x)\le 3\qquad\textbf{(C) }0\le f(x)\le 3\qquad\textbf{(D) }-1\le f(x)\le 4\qquad\textbf{(E) }\text{None of the above} $

2012 AIME Problems, 10

Tags: floor function , AMC

Find the number of positive integers $n$ less than $1000$ for which there exists a positive real number $x$ such that $n = x \lfloor x \rfloor$. [b]Note[/b]: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

2015 Estonia Team Selection Test, 2

Tags: rectangle , square , combinatorial geometry , combinatorics

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

2016 IFYM, Sozopol, 3

Tags: combinatorial geometry , combinatorics

Let $A_1 A_2…A_{66}$ be a convex 66-gon. What’s the greatest number of pentagons $A_i A_{i+1} A_{i+2} A_{i+3} A_{i+4},1\leq i\leq 66,$ which have an inscribed circle? ($A_{66+i}\equiv A_i$).

2011 Federal Competition For Advanced Students, Part 2, 2

Tags: combinatorics proposed , combinatorics

We consider permutations $f$ of the set $\mathbb{N}$ of non-negative integers, i.e. bijective maps $f$ from $\mathbb{N}$ to $\mathbb{N}$, with the following additional properties: \[f(f(x)) = x \quad \mbox{and}\quad \left| f(x)-x\right| \leqslant 3\quad\mbox{for all } x \in\mathbb{N}\mbox{.}\] Further, for all integers $n > 42$, \[\left.M(n)=\frac{1}{n+1}\sum_{j=0}^n \left|f(j)-j\right|<2,011\mbox{.}\right.\] Show that there are infinitely many natural numbers $K$ such that $f$ maps the set \[\left\{ n\mid 0\leqslant n\leqslant K\right\}\] onto itself.

2023 China Team Selection Test, P20

Tags: polynomial , number theory

Let $a,b,d$ be integers such that $\left|a\right| \geqslant 2$, $d \geqslant 0$ and $b \geqslant \left( \left|a\right| + 1\right)^{d + 1}$. For a real coefficient polynomial $f$ of degree $d$ and integer $n$, let $r_n$ denote the residue of $\left[ f(n) \cdot a^n \right]$ mod $b$. If $\left \{ r_n \right \}$ is eventually periodic, prove that all the coefficients of $f$ are rational.

2012 India Regional Mathematical Olympiad, 1

Tags: geometry

Let ABCD be a unit square. Draw a quadrant of a circle with A as centre and B;D as end points of the arc. Similarly, draw a quadrant of a circle with B as centre and A;C as end points of the arc. Inscribe a circle ? touching the arc AC internally, the arc BD internally and also touching the side AB. Find the radius of the circle ?.

2018 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , polynomial , abstract algebra

$f(x)$ is polynomial with integer coefficients, with module not exceeded $5*10^6$. $f(x)=nx$ has integer root for $n=1,2,...,20$. Prove that $f(0)=0$

2011 AMC 10, 18

Tags: geometry , rectangle , trigonometry , trig identities , Law of Sines , AMC , angles

Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

1989 National High School Mathematics League, 14

Tags: geometry , 3D geometry , pyramid

In regular triangular pyramid $S-ABC$, hieght $SO=3$, length of sides of bottom surface is $6$. Projection of $A$ on plane $SBC$ is $O'$. $P\in AO',\frac{AP}{PO'}=8$. Draw a plane parallel to plane $ABC$ and passes $P$. Find the area of the cross section.

1947 Moscow Mathematical Olympiad, 128

Tags: algebra , coefficients , polynomial

Find the coefficient of $x^2$ after expansion and collecting the terms of the following expression (there are $k$ pairs of parentheses): $$((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2$$

2002 Tuymaada Olympiad, 1

Tags: limit , number theory proposed , number theory

A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite. [i]Proposed by A. Golovanov[/i]

1978 AMC 12/AHSME, 17

Tags: function , AMC

If $k$ is a positive number and $f$ is a function such that, for every positive number $x$, \[\left[f(x^2+1)\right]^{\sqrt{x}}=k;\] then, for every positive number $y$, \[\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}\] is equal to $\textbf{(A) }\sqrt{k}\qquad\textbf{(B) }2k\qquad\textbf{(C) }k\sqrt{k}\qquad\textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}$

2014-2015 SDML (High School), 3

Tags: 2014-15 SDML Round 2 , SDML High School

A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after $671$ seconds? $\text{(A) }148\qquad\text{(B) }154\qquad\text{(C) }167\qquad\text{(D) }217\qquad\text{(E) }223$