Olympiad problems

2021 CIIM, 2

Tags: algebra

Let $r>s$ be positive integers. Let $P(x)$ and $Q(x)$ be distinct polynomials with real coefficients, non-constant(s), such that $P(x)^r-P(x)^s=Q(x)^r-Q(x)^s$ for every $x\in \mathbb{R}$. Prove that $(r,s)=(2,1)$.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.4

Tags: geometry , 3d geometry

The wire is bent in the form of a square with side $2$. Find the volume of the body consisting of all points in space located at a distance not exceeding $1$ from at least one point of the wire.

2000 AMC 8, 20

Tags:

You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $\$1.02$, with at least one coin of each type. How many dimes must you have? $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

2018 Dutch BxMO TST, 1

Tags: coloring , circles , combinatorics

We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours.

2000 Cono Sur Olympiad, 3

Tags: rectangle , rotation , geometry

Inside a $2\times 2$ square, lines parallel to a side of the square (both horizontal and vertical) are drawn thereby dividing the square into rectangles. The rectangles are alternately colored black and white like a chessboard. Prove that if the total area of the white rectangles is equal to the total area of the black rectangles, then one can cut out the black rectangles and reassemble them into a $1\times 2$ rectangle.

2014 Ukraine Team Selection Test, 9

Tags: diophantine equation , number theory , prime number , parameterization , diophantine , system of equations

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

2021 EGMO, 2

Tags: functional equation

Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation \[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$. Here, $\mathbb{Q}$ denotes the set of rational numbers.

2007 CHKMO, 3

Tags: circumcircle , symmetry , power of a point , radical axis , geometry

A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.

2007 Stanford Mathematics Tournament, 1

Tags:

Find all real roots of $f$ if $f(x^{1/9})=x^2-3x-4$.

2016 Online Math Open Problems, 4

Tags:

Let $G=10^{10^{100}}$ (a.k.a. a googolplex). Then \[\log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G\] can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine the sum of the digits of $m+n$. [i]Proposed by Yannick Yao[/i]

2016 BMT Spring, 8

Tags: combinatorics

How many ways are there to divide $10$ candies between $3$ Berkeley students and $4$ Stanford students, if each Berkeley student must get at least one candy? All students are distinguishable from each other; all candies are indistinguishable.

2013 Federal Competition For Advanced Students, Part 2, 6

Tags: 3d geometry , octahedron , sphere , geometry

Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$. Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.

2019 AIME Problems, 15

Tags: geometry , symmedian , radical axis

Let $\overline{AB}$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $\overline{AB}$. Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$. Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$. Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. Line $PQ$ intersects $\omega$ at $X$ and $Y$. Assume that $AP=5$, $PB=3$, $XY=11$, and $PQ^2 = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2020 BAMO, D/2

Tags: asymptote

Here’s a screenshot of the problem. If someone could LaTEX a diagram, that would be great!

1941 Moscow Mathematical Olympiad, 072

Tags: number theory , divisible

Find the number $\overline {523abc}$ divisible by $7, 8$ and $9$.

1972 Putnam, B3

Tags: group theory

A group $G$ has elements $g,h$ satisfying $ghg=hg^{2}h, g^{3}=1$ and $h^n=1$ for some odd integer $n$. Prove that $h=e$, where $e$ is the identity element.

2021 Alibaba Global Math Competition, 17

Tags: algebra , polynomial , number theory , finite field

Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by \[\tau(f)(x)=f(x+1).\] Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$.

2024 BMT, 3

Tags: geometry

A square with side length $6$ has a circle with radius $2$ inside of it, with the centers of the square and circle vertically aligned. Aarush is standing $4$ units directly above the center of the circle, at point $P.$ What is the area of the region inside the square that he can see? (Assume that Aarush can only see parts of the square along straight lines of sight from $P$ that are unblocked by any other objects.) [center] [img] https://cdn.artofproblemsolving.com/attachments/7/7/ede4444ab82235fc90a27c0b481d320b486cf2.png [/img] [/center]

2013 NIMO Summer Contest, 14

Tags:

Let $p$, $q$, and $r$ be primes satisfying \[ pqr = 189999999999999999999999999999999999999999999999999999962. \] Compute $S(p) + S(q) + S(r) - S(pqr)$, where $S(n)$ denote the sum of the decimals digits of $n$. [i]Proposed by Evan Chen[/i]

1978 USAMO, 3

Tags: induction , algebra , equation , additive number theory

An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.

2013 Abels Math Contest (Norwegian MO) Final, 3

Tags: number theory , prime , divide

A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

2019 CCA Math Bonanza, I15

Tags: poor eyesight

Before Harry Potter died, he decided to bury his wand in one of eight possible locations (uniformly at random). A squad of Death Eaters decided to go hunting for the wand. They know the eight locations but have poor vision, so even if they're at the correct location they only have a $50\%$ chance of seeing the wand. They also get tired easily, so they can only check three different locations a day. At least they have one thing going for them: they're clever. Assuming they strategize optimally, what is the expected number of days it will take for them to find the wand? [i]2019 CCA Math Bonanza Individual Round #15[/i]

2011 Brazil National Olympiad, 4

Tags: number theory , number theory gcd , induction

Do there exist $2011$ positive integers $a_1 < a_2 < \ldots < a_{2011}$ such that $\gcd(a_i,a_j) = a_j - a_i$ for any $i$, $j$ such that $1 \le i < j \le 2011$?

1962 AMC 12/AHSME, 32

Tags: induction , arithmetic series

If $ x_{k\plus{}1} \equal{} x_k \plus{} \frac12$ for $ k\equal{}1, 2, \dots, n\minus{}1$ and $ x_1\equal{}1,$ find $ x_1 \plus{} x_2 \plus{} \dots \plus{} x_n.$ $ \textbf{(A)}\ \frac{n\plus{}1}{2} \qquad \textbf{(B)}\ \frac{n\plus{}3}{2} \qquad \textbf{(C)}\ \frac{n^2\minus{}1}{2} \qquad \textbf{(D)}\ \frac{n^2\plus{}n}{4} \qquad \textbf{(E)}\ \frac{n^2\plus{}3n}{4}$

2010 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , divisibility , odd

Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$