Problem of the Days

"Days" since no regular schedule.

January 29, 2022 (Graham-Pollak theorem)

Determine the minimum number of complete bipartite graphs into which the edges of an $n$-vertex complete graph may be partitioned.

October 18, 2021 (KoMaL A.704)

Let $n$ be a nonnegative integer and let $T_n$ be the set of all points $(x,y,z)$ in space such that $x$, $y$, $z$ are nonnegative integers satisfying $x+y+z=n$. For which $n$ can $T_n$ be partitioned into triplets of points which are the vertices of an equilateral triangle of side length $\sqrt2$?

July 10, 2021 (Turkey TST 2000/6)

Suppose $f:\mathbb R\to\mathbb R$ is a function such that \[|f(x+y)-f(x)-f(y)|\le 1\quad\text{for all }x,y\in\mathbb R.\] Prove that there is a function $g:\mathbb R\to\mathbb R$ such that $|f(x)-g(x)|\le1$ and $g(x+y)=g(x)+g(y)$ for all $x,y\in\mathbb R$.

July 9, 2021 (Alon-Spencer)

There are $4n$ beads in a circle, four beads of each of $n$ colors. Show that we can choose a bead of each color so that no two chosen beads are adjacent.

July 8, 2021 (AMM 121.7.610)

Let $f:\mathbb R^2\to\mathbb R$ be a function such that for any unit square $ABCD$, \[f(A)+f(B)+f(C)+f(D)=0.\] Prove that $f(P)=0$ for every point $P$.

June 29, 2021 (arXiv 1509.04354)

Let $p$ be a prime number and let $A$ and $B$ be sets of residues modulo $p$ such that for all $a\in A$ and $b\in B$, $a+b$ is a nonzero quadratic residue modulo $p$. Prove that $|A|\cdot |B|\le p$.

May 28, 2020 (IMO 2016/3)

Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1$, $A_2$, $\ldots$, $A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.

February 23, 2020 (Taiwan TST 2016/2/Q2)

Find all functions $f:\mathbb Z\to\mathbb Z$ such that \[f(f(x)+f(y))+f(x)f(y)=f(x+y)f(x-y)\] for all integers $x$ and $y$.

December 9, 2019 (USA TST 2019/2)

Let $\mathbb Z/n\mathbb Z$ denote the set of integers considered modulo $n$ (hence $\mathbb Z/n\mathbb Z$ has $n$ elements). Find all positive integers $n$ for which there exists a bijective function $g:\mathbb Z/n\mathbb Z\to\mathbb Z/n\mathbb Z$, such that the 101 functions \[g(x),\quad g(x)+x,\quad g(x)+2x,\quad\ldots,\quad g(x)+100x\] are all bijections on $\mathbb Z/n\mathbb Z$.

October 19, 2019 (Ireland 1991/9)

Find all functions $f:\mathbb Q_{>0}\to\mathbb Q_{>0}$ such that

1. $f(x)+f(1/x)=1$ and
2. $f(2x)=2f(f(x))$ for all $x\in\mathbb Q_{>0}$.

September 25, 2019 (USAMO 2010/6)

A blackboard contains $68$ pairs of nonzero integers. Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the $136$ integers, subject to the condition that no two erased integers may add to $0$. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which $68$ pairs have been written on the board.

September 15, 2019 (Unknown)

Find a binary operation $\mathbin\diamondsuit$, such that by composing $\mathbin\diamondsuit$, it is possible to evaluate $a+b$, $a-b$, $a\times b$, and $a\div b$ in finitely many steps for all nonzero real numbers $a$ and $b$, or show that none exists. Fixed constants, such as $-4.5$ and $\pi$, are allowed.

August 27, 2019 (Classic)

Let $a$ and $b$ be positive integers with $a>b$. We are given a group of $N$ people. First they are split into $a$ groups, such that each group contains at least one person. Then they are split into $b$ groups, such that each group contains at least one person, independently of how they were split initially. Must there be a person who ended up in a strictly larger group?

July 28, 2019 (ELMO 2018/3)

Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane.

1. Can Evan construct${}^*$ the reflection of $A$ over $\ell$?
2. Can Evan construct the foot of the altitude from $A$ to $\ell$?

${}^*$To construct a point, Evan must have an algorithm which marks the point in finitely many steps.

Proposed by Zack Chroman.

July 23, 2019 (IMO 2017/2)

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\rightarrow\mathbb R$ such that, for any real numbers $x$ and $y$, \[f(f(x)f(y))+f(x+y)=f(xy).\]

Proposed by Dorlir Ahmeti.

July 20, 2019 (Sharygin 2014)

Let $ABCD$ be a cyclic quadrilateral. Prove that if the symmedian points of $\triangle ABC$, $\triangle BCD$, $\triangle CDA$, and $\triangle DAB$ are concyclic, then quadrilateral $ABCD$ has two parallel sides.

Proposed by Nikolai Beluhov.

July 18, 2019 (ISL 2018/N5)

Positive integers $x$, $y$, $z$, $t$ satisfy $xy-zt=x+y=z+t$. Can $xy$ and $zt$ both be perfect squares?

July 11, 2019 (India TST 2019/8)

Let $ABC$ be an acute triangle with circumcircle $\Gamma$ and altitudes $\overline{AD}$, $\overline{BE}$, $\overline{CF}$ meeting at $H$. Let $\omega$ be the circumcircle of $\triangle DEF$. Point $S\ne A$ lies on $\Gamma$ such that $DS=DA$. Line $\overline{AD}$ meets $\overline{EF}$ at $Q$, and meets $\omega$ at $L\ne D$. Point $M$ is chosen such that $\overline{DM}$ is a diameter of $\omega$. Point $P$ lies on $\overline{EF}$ with $\overline{DP}\perp\overline{EF}$. Prove that lines $SH$, $MQ$, $PL$ are concurrent.

Proposed by Anant Mudgal.