Olympiad problems

2006 Taiwan National Olympiad, 1

Tags: inequalities

Positive reals $a,b,c$ satisfy $abc=1$. Prove that $\displaystyle 1+ \frac{3}{a+b+c} \ge \frac{6}{ab+bc+ca}$.

2005 Taiwan TST Round 3, 1

Tags: function , number theory , algebra , divisibility , functional equation

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

2008 Harvard-MIT Mathematics Tournament, 28

Tags: 3d geometry , octahedron , geometry

Let $ P$ be a polyhedron where every face is a regular polygon, and every edge has length $ 1$. Each vertex of $ P$ is incident to two regular hexagons and one square. Choose a vertex $ V$ of the polyhedron. Find the volume of the set of all points contained in $ P$ that are closer to $ V$ than to any other vertex.

2021 Science ON all problems, 2

Tags: counting , combinatorics

Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$. [i] (Vlad Robu) [/i]

1997 AMC 12/AHSME, 11

Tags:

In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $ 23$, $ 14$, $ 11$, and $ 20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $ 18$, what is the least number of points she could have scored in the tenth game? $ \textbf{(A)}\ 26\qquad \textbf{(B)}\ 27\qquad \textbf{(C)}\ 28\qquad \textbf{(D)}\ 29\qquad \textbf{(E)}\ 30$

the 14th XMO, P4

Tags: grid , combinatorics , modular arithmetic

In an $n$ by $n$ grid, each cell is filled with an integer between $1$ and $6$. The outmost cells all contain the number $1$, and any two cells that share a vertex has difference not equal to $3$. For any vertex $P$ inside the grid (not including the boundary), there are $4$ cells that have $P$ has a vertex. If these four cells have exactly three distinct numbers $i$, $j$, $k$ (two cells have the same number), and the two cells with the same number have a common side, we call $P$ an $ijk$-type vertex. Let there be $A_{ijk}$ vertices that are $ijk$-type. Prove that $A_{123}\equiv A_{246} \pmod 2$.

2015 India IMO Training Camp, 2

Tags: number theory , prime

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

1991 Swedish Mathematical Competition, 5

Tags: digit , number theory

Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.

2021 Purple Comet Problems, 18

Tags: algebra , polynomial , geometry

The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle.

1951 AMC 12/AHSME, 29

Tags: ratio

Of the following sets of data the only one that does not determine the shape of a triangle is: $ \textbf{(A)}\ \text{the ratio of two sides and the included angle} \\ \qquad\textbf{(B)}\ \text{the ratios of the three altitudes} \\ \qquad\textbf{(C)}\ \text{the ratios of the three medians} \\ \qquad\textbf{(D)}\ \text{the ratio of the altitude to the corresponding base} \\ \qquad\textbf{(E)}\ \text{two angles}$

2016 Saudi Arabia BMO TST, 1

Tags: polynomial , quadratic polynomial , quadratic , algebra

Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.

1960 AMC 12/AHSME, 35

Tags: geometry , search

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values: $ \textbf{(A)}\ \text{zero} \qquad\textbf{(B)}\ \text{one} \qquad\textbf{(C)}\ \text{two} \qquad\textbf{(D)}\ \text{three} \qquad\textbf{(E)}\ \text{infinitely many} $

1997 Junior Balkan MO, 1

Tags: combinatorics , geometry , pigeonhole principle , inequalities , symmetry , area of a triangle

Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $\frac 18$. [i]Bulgaria[/i]

1990 National High School Mathematics League, 6

Tags: ellipse , conic

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ passes point $(2,1)$, then all points $(x,y)$ on the ellipse that $|y|>1$ are (shown as shadow) [img]https://graph.baidu.com/resource/122481219e60931bb707101582696834.jpg[/img]

1984 Tournament Of Towns, (053) O1

Tags: combinatorics , algebra

The price of $175$ Humpties is more than the price of $125$ Dumpties but less than that of $126$ Dumpties. Prove that you cannot buy three Humpties and one Dumpty for (a) $80$ cents. (b) $1$ dollar. (S Fomin, Leningrad) PS. (a) for Juniors , (a),(b) for Seniors

2021 Regional Competition For Advanced Students, 4

Tags: number theory , divide , divisible

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

2024 All-Russian Olympiad, 5

Tags: combinatorics

A straight road consists of green and red segments in alternating colours, the first and last segment being green. Suppose that the lengths of all segments are more than a centimeter and less than a meter, and that the length of each subsequent segment is larger than the previous one. A grasshopper wants to jump forward along the road along these segments, stepping on each green segment at least once an without stepping on any red segment (or the border between neighboring segments). Prove that the grasshopper can do this in such a way that among the lengths of his jumps no more than $8$ different values occur. [i]Proposed by T. Korotchenko[/i]

1991 Federal Competition For Advanced Students, 1

Tags: number theory

Suppose that $ a,b,$ and $ \sqrt[3]{a}\plus{}\sqrt[3]{b}$ are rational numbers. Prove that $ \sqrt[3]{a}$ and $ \sqrt[3]{b}$ are also rational.

2022 SAFEST Olympiad, 2

Tags: integer , number theory , divisibility

Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

2012 Korea National Olympiad, 1

Tags: circumcircle , trigonometry , rectangle , trapezoid , geometry

Let $ ABC $ be an obtuse triangle with $ \angle A > 90^{\circ} $. Let circle $ O $ be the circumcircle of $ ABC $. $ D $ is a point lying on segment $ AB $ such that $ AD = AC $. Let $ AK $ be the diameter of circle $ O $. Two lines $ AK $ and $ CD $ meet at $ L $. A circle passing through $ D, K, L $ meets with circle $ O $ at $ P ( \ne K ) $ . Given that $ AK = 2, \angle BCD = \angle BAP = 10^{\circ} $, prove that $ DP = \sin ( \frac{ \angle A}{2} )$.

2020 HMNT (HMMO), 3

Tags: combinatorics

Jody has $6$ distinguishable balls and $6$ distinguishable sticks, all of the same length. How many ways are there to use the sticks to connect the balls so that two disjoint non-interlocking triangles are formed? Consider rotations and reflections of the same arrangement to be indistinguishable.

2006 All-Russian Olympiad, 8

Tags: ceiling function , function , combinatorics , probabilistic method

At a tourist camp, each person has at least $50$ and at most $100$ friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) $1331$ different colors, and any person has $20$ friends whose t-shirts all have pairwisely different colors.

2022 Stanford Mathematics Tournament, 2

Tags:

Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$. Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$.

Kyiv City MO Juniors 2003+ geometry, 2016.8.51

Tags: geometry , equal angles , equal segments

In the quadrilateral $ABCD$, shown in fig. , the equations are true: $\angle ABC = \angle BCD$ and $2AB = CD$. On the side $BC$, a point $X$ is selected such that $\angle BAX = \angle CDA$. Prove that $AX = AD$. [img]https://cdn.artofproblemsolving.com/attachments/2/9/0884eb311d1e40300c1e5980fd53eaadfa7a25.png[/img]

2023 South East Mathematical Olympiad, 2

Tags: complex number , algebra

For a non-empty finite complex number set $A$, define the "[i]Tao root[/i]" of $A$ as $\left|\sum_{z\in A} z \right|$. Given the integer $n\ge 3$, let the set $$U_n = \{\cos\frac{2k \pi}{n}+ i\sin\frac{2k \pi}{n}|k=0,1,...,n-1\}.$$Let $a_n$ be the number of non-empty subsets in which the [i]Tao root [/i] of $U_n$ is $0$ , $b_n$ is the number of non-empty subsets of $U_n$ whose [i]Tao root[/i] is $1$. Compare the sizes of $na_n$ and $2b_n$.