Olympiad problems

1996 All-Russian Olympiad Regional Round, 9.5

Find all natural numbers that have exactly six divisors whose sum is $3500$.

1998 Harvard-MIT Mathematics Tournament, 6

Tags: calculus

Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place $1,875$ feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of $100 \dfrac{\text{ft}}{\text{s}}$. At the same time, Edward started running away with an acceleration of $1 \dfrac{\text{ft}}{\text{s}^2}$. Assuming that air resistance causes the arrow to decelerate at $1 \dfrac{\text{ft}}{\text{s}^2}$, and that it does hit Edward, how fast was the arrow moving at the moment of impact (in $\dfrac{\text{ft}}{\text{s}}$)?

2019 Jozsef Wildt International Math Competition, W. 5

Tags: matrix

Let $n \geq 1$. Find a set of distincts real numbers $\left(x_j\right)_{1\leq j\leq n}$ such that for any bijections $f : \{1, 2,\cdots ,n\}^2 \to \{1, 2,\cdots ,n\}^2$ the matrix $\left(x_{f(i,j)}\right)_{1\leq i,j\leq n}$ is invertible.

1967 IMO Longlists, 43

Tags: algebra , polynomial , diophantine equation , parametric equation , root

The equation \[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\] is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$

2014 Paraguay Mathematical Olympiad, 2

Tags: number theory

Clau writes all four-digit natural numbers where $3$ and $7$ are always together. How many digits does she write in total?

1957 Putnam, A3

Tags: trigonometry , inequalities , cosine

Let $a,b$ be real numbers and $k$ a positive integer. Show that $$ \left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1$$ whenever the left side is defined.

2009 Flanders Math Olympiad, 2

Tags: number theory , divisor

A natural number has four natural divisors: $1$, the number itself, and two real divisors. That number plus $9$ is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.

2005 AIME Problems, 1

Tags: geometry , floor function

Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$.

2023 CUBRMC, 5

Tags: algebra

The quadratic polynomial $f(x)$ has the expansion $2x^2 - 3x + r$. What is the largest real value of $r$ for which the ranges of the functions $f(x)$ and $f(f(x))$ are the same set?

1906 Eotvos Mathematical Competition, 3

Tags: number theory , even , permutation

Let $a_1, a_2, ...,a_n$ represent an arbitrary arrangement of the numbers $1, 2, ...,n$. Prove that, if $n$ is odd, the product $$(a_1 - 1)(a_2 - 2) ... (a_n -n)$$ is an even number.

2022 Moldova Team Selection Test, 2

Tags: inequalities

Real numbers $a, b, c, d$ satisfy $$a^2+b^2+c^2+d^2=4.$$ Find the greatest possible value of $$E(a,b,c,d)=a^4+b^4+c^4+d^4+4(a+b+c+d)^2 .$$

2018 Czech-Polish-Slovak Junior Match, 6

Tags: inequalities , algebra

Positive real numbers $a, b$ are such that $a^3 + b^3 = 2$. Show that that $\frac{1}{a}+\frac{1}{b}\ge 2(a^2 - a + 1)(b^2 - b + 1)$.

2014 Romania Team Selection Test, 4

Tags: induction , modular arithmetic , relatively prime , prime factorization , number theory

Let $k$ be a nonzero natural number and $m$ an odd natural number . Prove that there exist a natural number $n$ such that the number $m^n+n^m$ has at least $k$ distinct prime factors.

2011 Croatia Team Selection Test, 4

Tags: induction , relatively prime , number theory

We define the sequence $x_n$ so that \[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\] Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.

2023 HMNT, 28

Tags:

There is a unique quadruple of positive integers $(a,b,c,k)$ such that $c$ is not a perfect square and $a+\sqrt{b+\sqrt{c}}$ is a root of the polynomial $x^4-20x^3+108x^2-kx+9.$ Compute $c.$

2014 ELMO Shortlist, 5

Tags: inequalities , induction , number theory

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

2010 F = Ma, 14

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A $\text{5.0 kg}$ block with a speed of $\text{8.0 m/s}$ travels $\text{2.0 m}$ along a horizontal surface where it makes a head-on, perfectly elastic collision with a $\text{15.0 kg}$ block which is at rest. The coefficient of kinetic friction between both blocks and the surface is $0.35$. How far does the $\text{15.0 kg}$ block travel before coming to rest? (A) $\text{0.76 m}$ (B) $\text{1.79 m}$ (C) $\text{2.29 m}$ (D) $\text{3.04 m}$ (E) $\text{9.14 m}$

Novosibirsk Oral Geo Oly VIII, 2016.5

Tags: geometry , parallelogram , perpendicular

In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]

2020 IMO Shortlist, C2

Tags: coloring , combinatorics , quadrilateral , partition

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

2001 AMC 8, 14

Tags:

Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose? - Meat: beef, chicken, pork - Vegetables: baked beans, corn, potatoes, tomatoes - Dessert: brownies, chocolate cake, chocolate pudding, ice cream $ \text{(A)}\ 4\qquad\text{(B)}\ 24\qquad\text{(C)}\ 72\qquad\text{(D)}\ 80\qquad\text{(E)}\ 144 $

2008 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , circumcircle

In a triangle $ ABC$, take point $ D$ on $ BC$ such that $ DB \equal{} 14, DA \equal{} 13, DC \equal{} 4$, and the circumcircle of $ ADB$ is congruent to the circumcircle of $ ADC$. What is the area of triangle $ ABC$?

2025 Kyiv City MO Round 2, Problem 2

Tags: algebra , combinatorics

For some positive integer $ n $, Katya wrote the numbers from $ 1 $ to $ 2^n $ in a row in increasing order. Oleksii rearranged Katya's numbers and wrote the new sequence directly below the first row. Then, they calculated the sum of the two numbers in each column. Katya calculated $ N $, the number of powers of two among the results, while Oleksii calculated $ K $, the number of distinct powers of two among the results. What is the maximum possible value of $ N + K $? [i]Proposed by Oleksii Masalitin[/i]

2005 Oral Moscow Geometry Olympiad, 6

Tags: midpoint , geometry , incenter , perpendicular

Let $A_1,B_1,C_1$ are the midpoints of the sides of the triangle $ABC, I$ is the center of the circle inscribed in it. Let $C_2$ be the intersection point of lines $C_1 I$ and $A_1B_1$. Let $C_3$ be the intersection point of lines $CC_2$ and $AB$. Prove that line $IC_3$ is perpendicular to line $AB$. (A. Zaslavsky)

2007 Today's Calculation Of Integral, 224

Tags: calculus , integration , trigonometry , calculus computations

Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

2011 China National Olympiad, 3

Tags: modular arithmetic , algorithm , prime number , divisibility , number theory

Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]