Olympiad problems

2001 Putnam, 1

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Let $n$ be an even positive integer. Write the numbers $1, 2, \cdots, n^2$ in the squares of an $n \times n$ grid so that the $k$th row, from left to right, is \[ (k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n. \] Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.

2017 Serbia National Math Olympiad, 3

Tags: geometry , excircle

Let $k$ be the circumcircle of $\triangle ABC$ and let $k_a$ be A-excircle .Let the two common tangents of $k,k_a$ cut $BC$ in $P,Q$.Prove that $\measuredangle PAB=\measuredangle CAQ$.

2023 CMIMC Team, 13

Tags: team

Suppose that the sequence of real numbers $a_1,a_2,\ldots$ satisfies $a_1 = - \sqrt{1}, a_2 = \sqrt{2}$, and for all $k > 1$, \[ \frac{a_{k+1}+a_{k-1}}{a_k} = \frac{\sqrt{3} + \sqrt{1}}{\sqrt{2}}. \] Find $a_{2023}$. [i]Proposed by Kevin You[/i]

1997 Akdeniz University MO, 2

Tags: inequalities , algebra

Let $x,y,z,t$ be real numbers such that, $1 \leq x \leq y \leq z \leq t \leq 100$. Find minimum value of $$\frac{x}{y}+\frac{z}{t}$$

2006 China National Olympiad, 1

Tags: induction , inequalities

Let $a_1,a_2,\ldots,a_k$ be real numbers and $a_1+a_2+\ldots+a_k=0$. Prove that \[ \max_{1\leq i \leq k} a_i^2 \leq \frac{k}{3} \left( (a_1-a_2)^2+(a_2-a_3)^2+\cdots +(a_{k-1}-a_k)^2\right). \]

Geometry Mathley 2011-12, 16.4

Tags: geometry , orthocenter , incenter , circumcenter

A triangle $ABC$ is inscribed in the circle $(O)$, and has incircle $(I)$. The circles with diameter $IA$ meets $(O)$ at $A_1$ distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Line $B_1C_1$ meets $BC$ at $A_2$, and points $B_2,C_2$ are defined in the same manner. Prove that $O$ is the orthocenter of triangle $A_2B_2C_2$. Trần Minh Ngọc

2015 Spain Mathematical Olympiad, 1

Tags: polyhedron , graph coloring , combinatorics

All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is [i]Extremadura[/i] if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of [i]Extremadura[/i] faces of this polyhedron is always even?

OMMC POTM, 2023 6

Tags: combinatorics

Choose a permutation of$ \{1,2, ..., 20\}$ at random. Let $m$ be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of $2^m$. [i]Proposed by Evan Chang (squareman), USA[/i]

1997 AMC 8, 3

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Which of the following numbers is the largest? $\textbf{(A)}\ 0.97 \qquad \textbf{(B)}\ 0.979 \qquad \textbf{(C)}\ 0.9709 \qquad \textbf{(D)}\ 0.907 \qquad \textbf{(E)}\ 0.9089$

2016 CCA Math Bonanza, I8

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Let $f(x) = x^2 + x + 1$. Determine the ordered pair $(p,q)$ of primes satisfying $f(p) = f(q) + 242$. [i]2016 CCA Math Bonanza #8[/i]

2022 Estonia Team Selection Test, 2

Tags: number theory , first digit

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.

2016 India Regional Mathematical Olympiad, 5

Tags: number theory , sum of squares

a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples. b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.

2005 National Olympiad First Round, 31

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If the equation system \[\begin{array}{rcl} f(x) + g(x) &=& 0 \\ f(x)-(g(x))^3 &=& 0 \end{array}\] has more than one real roots, where $a,b,c,d$ are reals and $f(x)=x^2 + ax+b$, $g(x)=x^2 + cx + d$, at most how many distinct real roots can the equation $f(x)g(x) = 0$ have? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

2005 iTest, 7

Tags: binomial coefficient , algebra , binomial

Find the coefficient of the fourth term of the expansion of $(x+y)^{15}$.

1985 AMC 12/AHSME, 15

Tags: logarithm

If $ a$ and $ b$ are positive numbers such that $ a^b \equal{} b^a$ and $ b \equal{} 9a$, then the value of $ a$ is: $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \sqrt [9] {9}\qquad \textbf{(D)}\ \sqrt [3] {9}\qquad \textbf{(E)}\ \sqrt [4] {3}$

2014 BMT Spring, 3

Tags: number theory

Together, Abe and Bob have less than or equal to \$ $100$. When Corey asks them how much money they have, Abe says that the reciprocal of his money added to Bob’s money is thirteen times as much as the sum of Abe’s money and the reciprocal of Bob’s money. If Abe and Bob both have integer amounts of money, how many possible values are there for Abe’s money?

1992 China National Olympiad, 3

Tags: combinatorics

Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an [i]adjustment[/i] as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite [i]adjustments[/i].

1975 Polish MO Finals, 2

Tags: geometry , 3d geometry , tetrahedron , combinatorial geometry

On the surface of a regular tetrahedron of edge length $1$ are given finitely many segments such that every two vertices of the tetrahedron can be joined by a polygonal line consisting of given segments. Can the sum of the lengths of the given segments be less than $1+\sqrt3 $?

2016 Oral Moscow Geometry Olympiad, 5

Tags: geometry , excenter , concurrency

Points $I_A, I_B, I_C$ are the centers of the excircles of $ABC$ related to sides $BC, AC$ and $AB$ respectively. Perpendicular from $I_A$ to $AC$ intersects the perpendicular from $I_B$ to $B_C$ at point $X_C$. The points $X_A$ and $X_B$. Prove that the lines $I_AX_A, I_BX_B$ and $I_CX_C$ intersect at the same point.

2022 Dutch BxMO TST, 1

Tags: number theory , functional

Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.

2025 239 Open Mathematical Olympiad, 2

Tags: geometry

$AD$, $BE$, $CF$ are the heights of the acute—angled triangle $ABC$. A perpendicular is drawn to the segment $DE$ at point $E$. It intersects the height of $AD$ at point $G$. The point $J$ is chosen on the segment $BD$ in such a way that $BJ = CD$. The circumscribed circle of a triangle $BD$ intersects the segment $BE$ at point $Q$. Prove that the points $J$, $Q$, and $G$ are collinear.

2023 USA IMO Team Selection Test, 4

Tags: floor function , function , algebra

Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \] [i]Carl Schildkraut[/i]

1990 Tournament Of Towns, (278) 3

Tags: combinatorics , geometry , combinatorial geometry

A finite set $M$ of unit squares on the plane is considered. The sides of the squares are parallel to the coordinate axes and the squares are allowed to intersect. It is known that the distance between the centres of any pair of squares is no greater than $2$. Prove that there exists a unit square (not necessarily belonging to $M$) with sides parallel to the coordinate axes and which has at least one common point with each of the squares in $M$. (A Andjans, Riga)

2015 Germany Team Selection Test, 2

Tags: integer , positive , consecutive , form , number theory

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

1969 IMO Longlists, 8

Tags: function , algebra , functional equation , continuous function

Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.