Olympiad problems

2021-IMOC, A2

Tags: algebra , IMOC

For any positive integers $n$, find all $n$-tuples of complex numbers $(a_1,..., a_n)$ satisfying $$(x+a_1)(x+a_2)\cdots (x+a_n)=x^n+\binom{n}{1}a_1 x^{n-1}+\binom{n}{2}a_2^2 x^{n-2}+\cdots +\binom{n}{n-1} a_{n-1}^{n-1}+\binom{n}{n}a_n^n.$$ Proposed by USJL.

2011 Denmark MO - Mohr Contest, 5

Tags: number theory , Digits , last digits , Perfect Square

Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$. .

2003 AMC 10, 7

Tags: floor function

The symbolism $ \lfloor x\rfloor$ denotes the largest integer not exceeding $ x$. For example. $ \lfloor3\rfloor\equal{}3$, and $ \lfloor 9/2\rfloor\equal{}4$. Compute \[ \lfloor\sqrt1\rfloor\plus{}\lfloor\sqrt2\rfloor\plus{}\lfloor\sqrt3\rfloor\plus{}\cdots\plus{}\lfloor\sqrt{16}\rfloor. \]$ \textbf{(A)}\ 35 \qquad \textbf{(B)}\ 38 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 136$

2003 Federal Competition For Advanced Students, Part 1, 1

Tags: number theory proposed , number theory

Find all triples of prime numbers $(p, q, r)$ such that $p^q + p^r$ is a perfect square.

2024 CMI B.Sc. Entrance Exam, 4

Tags: inequalities , algebra

(a) For non negetive $a,b,c, r$ prove that \[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \] (b) Find an inequality for non negative $a,b,c$ with $a^4+b^4+c^4 + abc(a+b+c)$ on the greater side. (c) Prove that if $abc = 1$ for non negative $a,b,c$, $a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3$

2012 China Team Selection Test, 2

Tags: number theory proposed , number theory

For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if $\tau (m) < \tau (n)$ for all $m < n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.

1994 Romania TST for IMO, 4:

Tags: algebra unsolved , algebra

Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that $(1)$ $f(n) \leq n^8$ for any $n \geq 2$, $(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that $$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$

2013 All-Russian Olympiad, 2

Tags: geometry , circumcircle , geometric transformation , reflection , trigonometry , angle bisector , geometry proposed

Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$.

2012 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: equation , parameterization , algebra

Solve equation $$x^2-\sqrt{a-x}=a$$ where $x$ is real number and $a$ is real parameter

2021 LMT Fall, 3

Tags:

Two circles with radius $2$, $\omega_1$ and $\omega_2$, are centered at $O_1$ and $O_2$ respectively. The circles $\omega_1$ and $\omega_2$ are externally tangent to each other and internally tangent to a larger circle $\omega$ centered at $O$ at points $A$ and $B$, respectively. Let $M$ be the midpoint of minor arc $AB$. Let $P$ be the intersection of $\omega_1$ and $O_1M$, and let $Q$ be the intersection of $\omega_2$ and $O_2M$. Given that there is a point $R$ on $\omega$ such that $\triangle PQR$ is equilateral, the radius of $\omega$ can be written as $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers and $a$ and $c$ are relatively prime. Find $a+b+c$.

2017 Mathematical Talent Reward Programme, MCQ: P 10

Tags: differentiable functions , limit , calculus , function

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $\lim \limits_{x\to \infty}f'(x)=1$, then [list=1] [*] $f$ is increasing [*] $f$ is unbounded [*] $f'$ is bounded [*] All of these [/list]

1987 Greece National Olympiad, 3

Tags: algebra , Sequence , recurrence relation

There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$

2024 ELMO Shortlist, A2

Tags: combinatorics , Elmo , floor function

Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$ [i]Andrew Carratu[/i]

2024 Canada National Olympiad, 4

Tags: geometry , rectangle , combinatorics

Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le M$ and $1\le c\le d\le N$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. In terms of $M$ and $N$, what is the minimum $Q$ required to gaurantee to determine the location of the treasure?

2024 OMpD, 2

Tags: Matrices , algebra , linear algebra

Let $ n $ be a positive integer, and let $ A $ and $ B $ be $ n \times n $ matrices with real coefficients such that \[ ABBA - BAAB = A - B. \] (a) Prove that $ \text{Tr}(A) = \text{Tr}(B) $ and that $ \text{Tr}(A^2) = \text{Tr}(B^2) $. (b) If $BA^2B= A^2B^2$ and $AB^2A= B^2A^2$, prove that $ \det A = \det B $. Note: $ \text{Tr}(X) $ denotes the trace of $ X $, which is the sum of the elements on its main diagonal, and $ \det X $ denotes the determinant of $ X $.

2024 Baltic Way, 20

Tags: number theory , number theory proposed , Diophantine equation , Discriminant

Positive integers $a$, $b$ and $c$ satisfy the system of equations \begin{align*} (ab-1)^2&=c(a^2+b^2)+ab+1,\\ a^2+b^2&=c^2+ab. \end{align*} a) Prove that $c+1$ is a perfect square. b) Find all such triples $(a,b,c)$.

Durer Math Competition CD 1st Round - geometry, 2016.C+3

Tags: geometry , square , geometric inequality , Durer

Let $ABCD$ be a square with unit sides. Which interior point $P$ will the expression $\sqrt2 \cdot AP + BP + CP$ have a minimum value, and what is this minimum?

2011 AMC 12/AHSME, 6

Tags: AMC

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17 $

2023 AMC 12/AHSME, 25

Tags: AMC , AMC 12 , AMC 10 , 2023 AMC , 2023 AMC 10B , geometry

A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? $\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$

2010 Swedish Mathematical Competition, 5

Tags: angles , geometry , sidelengths

Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$. Determine the angles in the triangle for which the angle opposite to the side with the length $a$ is as big as possible.

2023 Harvard-MIT Mathematics Tournament, 7

Tags:

Let $ABC$ be a triangle. Point $D$ lies on segment $BC$ such that $\angle BAD = \angle DAC$. Point $X$ lies on the opposite side of line $BC$ as $A$ and satisfies $XB=XD$ and $\angle BXD = \angle ACB$. The point $Y$ is defined similarly. Prove that the lines $XY$ and $AD$ are perpendicular.

2021 Brazil National Olympiad, 3

Tags: number theory , floor function , powers , irrational number , Perfect Squares , Brazilian Math Olympiad , Brazil

Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is a perfect square minus $k$ for every integer $n$ with $n>N$.

2001 Stanford Mathematics Tournament, 3

Tags: Stanford , college , modular arithmetic

Find the 2000th positive integer that is not the diﬀerence between any two integer squares.

2020 CHKMO, 2

Tags: partition , number theory

Let $S={1,2,\ldots,100}$. Consider a partition of $S$ into $S_1,S_2,\ldots,S_n$ for some $n$, i.e. $S_i$ are nonempty, pairwise disjoint and $\displaystyle S=\bigcup_{i=1}^n S_i$. Let $a_i$ be the average of elements of the set $S_i$. Define the score of this partition by \[\dfrac{a_1+a_2+\ldots+a_n}{n}.\] Among all $n$ and partitions of $S$, determine the minimum possible score.

2006 Princeton University Math Competition, 3

Tags: algebra

Find the fifth root of $14348907$.