Olympiad problems

2011 Armenian Republican Olympiads, Problem 3

Tags: number theory

Find all integers $a, m, n, k,$ such that $(a^m+1)(a^n-1)=15^k.$

2001 Cuba MO, 1

Tags: combinatorics

In each square of a $3 \times 3$ board a real number is written. The element of the $i$ -th row and the $j$ -th column is equal to abso;uteof the difference of the sum of the elements of column $j$ and the sum of the elements of row $i$. Prove that every element of the board is equal to the sum or difference of two other elements on the board.

1997 AMC 12/AHSME, 22

Tags:

Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $ \$56$. The absolute difference between the amounts Ashley and Betty had to spend was $ \$19$. The absolute difference between the amounts Betty and Carlos had was $ \$7$, between Carlos and Dick was $ \$5$, between Dick and Elgin was $ \$4$, and between Elgin and Ashley was $ \$11$. How much did Elgin have? $ \textbf{(A)}\ \$6\qquad \textbf{(B)}\ \$7\qquad \textbf{(C)}\ \$8\qquad \textbf{(D)}\ \$9\qquad \textbf{(E)}\ \$10$

1999 Yugoslav Team Selection Test, Problem 1

Tags: polynomial , algebra

For a natural number $n$, let $P(x)$ be the polynomial of $2n$−th degree such that: $P(0) = 1$ and $P(k) = 2^{k-1}$ for $k = 1, 2, . . . , 2n$. Prove that $2P(2n + 1) - P(2n + 2) = 1$. P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution. :)

2010 Kurschak Competition, 3

Tags: number theory

For what positive integers $n$ and $k$ do there exits integers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_k$ such that the products $a_ib_j$ ($1\le i\le n,1\le j\le k$) give pairwise different residues modulo $nk$?

2014 EGMO, 3

Tags: quadratic , number theory , combinatorial number theory , divisor

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

1982 Vietnam National Olympiad, 3

Tags: 3d geometry , geometry

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.

2017 ELMO Shortlist, 2

Tags: number theory

An integer $n>2$ is called [i]tasty[/i] if for every ordered pair of positive integers $(a,b)$ with $a+b=n,$ at least one of $\frac{a}{b}$ and $\frac{b}{a}$ is a terminating decimal. Do there exist infinitely many tasty integers? [i]Proposed by Vincent Huang[/i]

2017 Harvard-MIT Mathematics Tournament, 16

Tags: algebra , complex number

Let $a$ and $b$ be complex numbers satisfying the two equations \begin{align*} a^3 - 3ab^2 & = 36 \\ b^3 - 3ba^2 & = 28i. \end{align*} Let $M$ be the maximum possible magnitude of $a$. Find all $a$ such that $|a| = M$.

Kyiv City MO Juniors Round2 2010+ geometry, 2021.8.2

Tags: geometry , incenter , right triangle

In a triangle $ABC$, $\angle B=90^o$ and $\angle A=60^o$, $I$ is the point of intersection of its angle bisectors. A line passing through the point $I$ parallel to the line $AC$, intersects the sides $AB$ and $BC$ at the points $P$ and $T$ respectively. Prove that $3PI+IT=AC$ . (Anton Trygub)

2012 Tournament of Towns, 2

Tags: number theory , prime , divisor

Let $C(n)$ be the number of prime divisors of a positive integer n. (For example, $C(10) = 2,C(11) = 1, C(12) = 2$). Consider set S of all pairs of positive integers $(a, b)$ such that $a\ne b$ and $C(a + b) = C(a) + C(b)$. Is set $S$ finite or infinite?

2012 NIMO Problems, 3

Tags:

Compute the sum of the distinct prime factors of $10101$. [i]Proposed by Lewis Chen[/i]

2014 Purple Comet Problems, 12

Tags: trigonometry , trapezoid , rectangle , geometry

The vertices of hexagon $ABCDEF$ lie on a circle. Sides $AB = CD = EF = 6$, and sides $BC = DE = F A = 10$. The area of the hexagon is $m\sqrt3$. Find $m$.

2011 May Olympiad, 4

Tags: combinatorics

Given $n$ points in a circle, Arnaldo write 0 or 1 in all the points. Bernado can do a operation, he can chosse some point and change its number and the numbers of the points on the right and left side of it. Arnaldo wins if Bernado can´t change all the numbers in the circle to 0, and Bernado wins if he can a) Show that Bernado can win if $n=101$ b) Show that Arnaldo wins if $n=102$

2007 Miklós Schweitzer, 4

Tags: abstract algebra , combinatorics

Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$-element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$'s (the empty sum is $0$). (translated by Miklós Maróti)

2015 Online Math Open Problems, 19

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Let $ABC$ be a triangle with $AB = 80, BC = 100, AC = 60$. Let $D, E, F$ lie on $BC, AC, AB$ such that $CD = 10, AE = 45, BF = 60$. Let $P$ be a point in the plane of triangle $ABC$. The minimum possible value of $AP+BP+CP+DP+EP+FP$ can be expressed in the form $\sqrt{x}+\sqrt{y}+\sqrt{z}$ for integers $x, y, z$. Find $x+y+z$. [i]Proposed by Yang Liu[/i]

2010 Moldova Team Selection Test, 1

Tags: polynomial , algebra

Let $ p\in\mathbb{R}_\plus{}$ and $ k\in\mathbb{R}_\plus{}$. The polynomial $ F(x)\equal{}x^4\plus{}a_3x^3\plus{}a_2x^2\plus{}a_1x\plus{}k^4$ with real coefficients has $ 4$ negative roots. Prove that $ F(p)\geq(p\plus{}k)^4$

LMT Team Rounds 2021+, A25 B26

Tags: number theory

Chandler the Octopus is making a concoction to create the perfect ink. He adds $1.2$ grams of melanin, $4.2$ grams of enzymes, and $6.6$ grams of polysaccharides. But Chandler accidentally added n grams of an extra ingredient to the concoction, Chemical $X$, to create glue. Given that Chemical $X$ contains none of the three aforementioned ingredients, and the percentages of melanin, enzymes, and polysaccharides in the final concoction are all integers, find the sum of all possible positive integer values of $n$. [i]Proposed by Taiki Aiba[/i]

2006 All-Russian Olympiad Regional Round, 10.4

Tags: algebra , polynomial , trinomial

Given $n > 1$ monic square trinomials $x^2 - a_1x + b_1$,$...$, $x^2-a_nx + b_n$, and all $2n$ numbers are $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n$ are different. Can it happen that each of the numbers $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n is the root of one of these trinomials?

2012 Oral Moscow Geometry Olympiad, 2

Tags: geometry , area , pentagon

In the convex pentagon $ABCDE$: $\angle A = \angle C = 90^o$, $AB = AE, BC = CD, AC = 1$. Find the area of the pentagon.

2005 AIME Problems, 14

Tags: geometry , analytic geometry , graphing lines , slope , rotation , trigonometry

Consider the points $A(0,12)$, $B(10,9)$, $C(8,0)$, and $D(-4,7)$. There is a unique square $S$ such that each of the four points is on a different side of $S$. Let $K$ be the area of $S$. Find the remainder when $10K$ is divided by $1000$.

2017 Balkan MO Shortlist, G1

Tags: geometry , circumcircle , fixed point

Let $ABC$ be an acute triangle. Variable points $E$ and $F$ are on sides $AC$ and $AB$ respectively such that $BC^2 = BA\cdot BF + CE \cdot CA$ . As $E$ and $F$ vary prove that the circumcircle of $AEF$ passes through a fixed point other than $A$ .

1976 AMC 12/AHSME, 7

Tags: function , algebra , polynomial

If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if $\textbf{(A) }|x|<1\qquad\textbf{(B) }|x|>1\qquad\textbf{(C) }x<-1\text{ or }-1<x<1\qquad$ $\textbf{(D) }x<1\qquad \textbf{(E) }x<-1$

1994 Baltic Way, 13

Tags: geometry

Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$, so that no two of the disks have a common interior point.

2021 Dutch IMO TST, 2

Tags: algebra , system of equations , system

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$