Olympiad problems

1988 Czech And Slovak Olympiad IIIA, 5

Tags: algebra , polynomial

Find all numbers $a \in (-2, 2)$ for which the polynomial $x^{154}-ax^{77}+1$ is a multiple of the polynomial $x^{14}-ax^{7}+1$.

2004 Poland - First Round, 3

Tags: trigonometry , vector , inequalities , calculus , derivative , geometry

3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and: $\frac{EF}{FD}=\frac{AD}{DB}$ Prove that $CF \bot AE$

2009 India Regional Mathematical Olympiad, 3

Tags: function , number theory

Show that $ 3^{2008} \plus{} 4^{2009}$ can be written as product of two positive integers each of which is larger than $ 2009^{182}$.

1980 IMO, 22

Tags: pascal triangle , modular arithmetic , number theory , representation

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

2022-2023 OMMC, 23

Tags:

Define the Fibonacci numbers such that $F_{1} = F_{2} = 1,$ $F_{k} = F_{k-1} + F_{k-2}$ for $k > 2.$ For large positive integers $n,$ the expression (containing $n$ nested square roots) $$\sqrt{2023 F^{2}_{2^{1}} + \sqrt{2023 F^{2}_{2^{2}} + \sqrt{2023 F_{2^{3}}^{2} \dots + \sqrt{2023 F^{2}_{2^{n}} }}}}$$ approaches $\frac{a + \sqrt{b}}{c}$ for positive integers $a,b,c$ where $\gcd(a,c) = 1.$ Find $a+b+c.$

2015 Azerbaijan JBMO TST, 2

Tags: combinatorics

$A=1\cdot4\cdot7\cdots2014$.Find the last non-zero digit of $A$ if it is known that $A\equiv 1\mod3$.

2003 Argentina National Olympiad, 2

Tags: combinatorics , number theory , algebra

On the blackboard are written the $2003$ integers from $1$ to $2003$. Lucas must delete $90$ numbers. Next, Mauro must choose $37$ from the numbers that remain written. If the $37$ numbers Mauro chooses form an arithmetic progression, Mauro wins. If not, Lucas wins. Decide if Lucas can choose the $90$ numbers he erases so that victory is assured.

2016 CCA Math Bonanza, L2.3

Tags: inequalities

Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $$AC^2+\left(3\cdot CD\right)^2+BC^2?$$ [i]2016 CCA Math Bonanza Lightning #2.3[/i]

2019 LIMIT Category B, Problem 5

Tags: geometry

A polygon has twice as many diagonals as it has sides. How many sides does it have?

2014 HMNT, 3

Tags: hmmt , inequalities , triangle inequality

The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of $42,$ and another is a multiple of $72$. What is the minimum possible length of the third side?

2024-IMOC, G2

Tags: geometry

Triangle $ABC$ has circumcenter $O$. $D$ is an arbitrary point on $BC$, and $AD$ intersects $\odot(ABC)$ at $E$. $S$ is a point on $\odot(ABC)$ such that $D, O, E, S$ are colinear. $AS$ intersects $BC$ at $P$. $Q$ is a point on $BC$ such that $D, O, A, Q$ are concylic. Prove that $\odot(ABC)$ is tangent to $\odot (APQ)$. [i]Proposed by chengbilly[/i]

2016 PAMO, 4

Tags: inequality , algebra

Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that $\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.

1990 AMC 12/AHSME, 13

Tags: arithmetic series

If the following instructions are carried out by a computer, which of $X$ will be printed because of instruction $5$? $1.$ Start $X$ at $3$ and $S$ at $0$ $2.$ Increase the value of $X$ by $2$. $3.$ Increase the value of $S$ by the value of $X$. $4.$ If $S$ is at least $10000$, then go to instsruction $5$; otherwise, go to instruction $2$ and proceed from there. $5.$ Print the value of $X$. $6.$ Stop. $\text{(A)} \ 19 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ 23 \qquad \text{(D)} \ 199 \qquad \text{(E)} \ 201$

2019 Turkey Junior National Olympiad, 1

Tags: gcd or factors , number theory , power of prime

Solve $2a^2+3a-44=3p^n$ in positive integers where $p$ is a prime.

2009 India IMO Training Camp, 6

Tags: function , combinatorics

Prove The Following identity: $ \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}$. The Second term on left hand side is to be regarded zero for j=0.

2013 Purple Comet Problems, 10

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Find the least positive integer $k$ so that the mean of the numbers $k,k + 1,k + 2,k + 3,\ldots,2k$ exceeds $200$.

2015 Indonesia MO Shortlist, G3

Tags: tangent , geometry , circumcircle , incircle

Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

2019 Belarusian National Olympiad, 9.7

Tags: algebra , polynomial

Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients such that $P(Q(x)^2)=P(x)\cdot Q(x)^2$. [i](I. Voronovich)[/i]

Champions Tournament Seniors - geometry, 2001.4

Tags: pentagon , equal angles , equal segments , geometry

Given a convex pentagon $ABCDE$ in which $\angle ABC = \angle AED = 90^o$, $\angle BAC= \angle DAE$. Let $K$ be the midpoint of the side $CD$, and $P$ the intersection point of lines $AD$ and $BK$, $Q$ be the intersection point of lines $AC$ and $EK$. Prove that $BQ = PE$.

2002 Bundeswettbewerb Mathematik, 1

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Planet Ypsilon has a calendar similar to ours: A year consists of $365$ days, and every month has $28$, $30$ or $31$ days. Prove that on Planet Ypsilon, a year must have $12$ months.

2022 Saudi Arabia BMO + EGMO TST, 2.1

Tags: geometry , circumcenter

Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB \parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ \parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circumcircle of triangle $PXQ$.

2002 Italy TST, 3

Tags: polynomial , vieta , relatively prime , number theory , algebra

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

1987 IMO Longlists, 18

Tags: geometric inequality , 3d geometry , polyhedron , geometry

Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality \[AF + AH + AC \leq AB + AD + AE + AG.\] In what cases does equality hold? [i]Proposed by France.[/i]

2018 Online Math Open Problems, 1

Tags:

Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards. Note: Two integers are relatively prime if no positive integer other than $1$ divides both numbers. [i]Proposed by ABCDE and Tristan Shin

1954 Moscow Mathematical Olympiad, 286

Tags: number theory , combinatorics , sum , 10-digit , subset

Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.