Olympiad problems

2011 Harvard-MIT Mathematics Tournament, 2

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A classroom has $30$ students and $30$ desks arranged in $5$ rows of $6$. If the class has $15$ boys and $15$ girls, in how many ways can the students be placed in the chairs such that no boy is sitting in front of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl?

2015 Romanian Master of Mathematics, 1

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

2024 HMNT, 5

Tags: team

Let $ABCD$ be a convex quadrilateral with area $202, AB = 4,$ and $\angle A = \angle B = 90^\circ$ such that there is exactly one point $E$ on line $CD$ satisfying $\angle AEB = 90^\circ.$ Compute the perimeter of $ABCD.$

2022 Korea Winter Program Practice Test, 1

Tags: diophantine equation , number theory

Prove that equation $y^2=x^3+7$ doesn't have any solution on integers.

2010 All-Russian Olympiad Regional Round, 9.1

Tags: quadratic , algebra , polynomial

Three quadratic polynomials $f_1(x) = x^2+2a_1x+b_1$, $f_2(x) = x^2+2a_2x+b_2$, $f_3(x) = x^2 + 2a_3x + b_3$ are such that $a_1a_2a_3 = b_1b_2b_3 > 1$. Prove that at least one polynomial has two distinct roots.

Oliforum Contest III 2012, 5

Tags: cyclic quadrilateral , orthogonal , geometry

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

2015 Princeton University Math Competition, B2

Tags: geometry

Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where $a$ and $b$ are positive coprime integers. What is $a + b$?

1989 Cono Sur Olympiad, 3

Tags: function

A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that: $f(n)=0$, if n is perfect $f(n)=0$, if the last digit of n is 4 $f(a.b)=f(a)+f(b)$ Find $f(1998)$

2022 JHMT HS, 2

Tags: factorial , number theory

Find the number of ordered pairs of positive integers $(m,n)$, where $m,n\leq 10$, such that $m!+n!$ is a multiple of $10$.

2009 Stanford Mathematics Tournament, 10

Tags: geometry

Right triangle $ABC$ is inscribed in circle $W$. $\angle{CAB}=65$ degrees, and $\angle{CBA}=25$ degrees. The median from $C$ to $AB$ intersects $W$ and line $D$. Line $l_1$ is drawn tangent to $W$ at $A$. Line $l_2$ is drawn tangent to $W$ at $D$. The lines $l_1$ and $l_2$ intersect at $P$ Determine $\angle{APD}$

2005 AMC 10, 18

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All of David's telephone numbers have the form $ 555\minus{}abc\minus{}defg$, where $ a$, $ b$, $ c$, $ d$, $ e$, $ f$, and $ g$ are distinct digits and in increasing order, and none is either $ 0$ or $ 1$. How many different telephone numbers can David have? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

2019 Thailand TST, 1

Tags: number theory

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

1979 VTRMC, 6

Tags: algebra

Suppose $a _ { n } > 0$ and $\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges. Determine whether $\sum _ { n = 1 } ^ { \infty } a _ { n } / S _ { n } ^ { 2 }$ converges, where $S _ { n } = a _ { 1 } + a _ { 2 } + \dots + a _ { n } .$

2013 Kosovo National Mathematical Olympiad, 4

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Let be $a,b,c$ three positive integer.Prove that $4$ divide $a^2+b^2+c^2$ only and only if $a,b,c$ are even.

1973 Chisinau City MO, 68

Tags: equilateral , similar , geometry

Inside the triangle $ABC$, point $O$ was chosen so that the triangles $AOB, BOC, COA$ turned out to be similar. Prove that triangle $ABC$ is equilateral.

2021 Brazil National Olympiad, 4

Tags: minimum value , combinatorics , algebra , square

A set $A$ of real numbers is framed when it is bounded and, for all $a, b \in A$, not necessarily distinct, $(a-b)^{2} \in A$. What is the smallest real number that belongs to some framed set?

2001 Putnam, 5

Tags: function

Let $a$ and $b$ be real numbers in the interval $\left(0,\tfrac{1}{2}\right)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.

2017 BMT Spring, 7

Tags: geometry

Determine the maximal area triangle such that all of its vertices satisfy $\frac{x^2}{9} + \frac{y^2}{16} = 1$.

2007 Turkey Junior National Olympiad, 3

Tags: number theory

Find all odd postive integers less than $2007$ such that the sum of all of its positive divisors is odd.

1994 Tournament Of Towns, (407) 5

Tags: similar , geometry , pentagon , combinatorial geometry

Does there exist a convex pentagon from which a similar pentagon can be cut off by a straight line? (S Tokarev)

2015 BMT Spring, 7

Tags: equal segments , geometry , equal angles

$X_1, X_2, . . . , X_{2015}$ are $2015$ points in the plane such that for all $1 \le i, j \le 2015$, the line segment $X_iX_{i+1} = X_jX_{j+1}$ and angle $\angle X_iX_{i+1}X_{i+2} = \angle X_jX_{j+1}X_{j+2}$ (with cyclic indices such that $X_{2016} = X_1$ and $X_{2017} = X_2$). Given fixed $X_1$ and $X_2$, determine the number of possible locations for $X_3$.

2022 Latvia Baltic Way TST, P11

Tags: circumcircle , geometry

Let $\triangle ABC$ be an acute triangle. Point $D$ is arbitrarily chosen on the side $BC$. Let the circumcircle of the triangle $\triangle ADB$ intersect the segment $AC$ at $M$, and the circumcircle of the triangle $\triangle ADC$ intersect the segment $AB$ at $N$. Prove that the tangents of the circumcircle of the triangle $\triangle AMN$ at $M$ and $N$ intersect at a point that lies on the line $BC$.

1989 AMC 12/AHSME, 25

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In a certain cross-country meet between two teams of five runners each, a runner who finishes in the $n^{th}$ position contributes $n$ to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 27 \qquad\textbf{(D)}\ 120 \qquad\textbf{(E)}\ 126 $

1967 IMO Longlists, 29

Tags: triangle , similar triangles , area of a triangle , geometry

$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

1915 Eotvos Mathematical Competition, 1

Tags: algebra , number theory , inequalities

Let $A, B, C$ be any three real numbers. Prove that there exists a number $\nu$ such that $$An^2 + Bn+ < n!$$ for every natural number $n > \nu.$