Olympiad problems

2021 Thailand TST, 3

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

2019 Jozsef Wildt International Math Competition, W. 1

Tags: pell equation , number theory , split method

The Pell numbers $P_n$ satisfy $P_0 = 0$, $P_1 = 1$, and $P_n=2P_{n-1}+P_{n-2}$ for $n\geq 2$. Find $$\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}$$

1987 AMC 8, 11

Tags:

The sum $2\frac17+3\frac12+5\frac{1}{19}$ is between $\text{(A)}\ 10\text{ and }10\frac12 \qquad \text{(B)}\ 10\frac12 \text{ and } 11 \qquad \text{(C)}\ 11\text{ and }11\frac12 \qquad \text{(D)}\ 11\frac12 \text{ and }12 \qquad \text{(E)}\ 12\text{ and }12\frac12$

2012 Princeton University Math Competition, B1

Tags: geometry

During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle $\theta$, in degrees, of the bottom of the cone when we look at it from the side? [img]https://cdn.artofproblemsolving.com/attachments/d/2/f8e3a7afb606dfd6fad277f547b116566a4a91.png[/img]

2000 Spain Mathematical Olympiad, 1

Tags: polynomial , parameterization , quadratic , algebra

Consider the polynomials \[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\] Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$

2019 Peru EGMO TST, 1

Tags: diophantine equation , diophantine , number theory , prime

Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.

2019 Yasinsky Geometry Olympiad, p3

Tags: geometry , angle bisector , circles , tangent circle

Two circles $\omega_1$ and $\omega_2$ are tangent externally at the point $P$. Through the point $A$ of the circle $\omega_1$ is drawn a tangent to this circle, which intersects the circle $\omega_2$ at points $B$ and $C$ (see figure). Line $CP$ intersects again the circle $\omega_1$ to $D$. Prove that the $PA$ is a bisector of the angle $DPB$. [img]https://1.bp.blogspot.com/-nmKZGdBXfao/XOd51gRFuyI/AAAAAAAAKO0/EYo2SCW0eGcJsF64-Avo6w73ugkIIQ30ACK4BGAYYCw/s1600/Yasinsky%2B2019%2Bp2.png[/img]

1962 Swedish Mathematical Competition, 5

Tags: geometry , 3d geometry , max , cube , tetrahedron

Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.

1986 IMO Longlists, 23

Tags: rectangle , geometry

Let $I$ and $J$ be the centers of the incircle and the excircle in the angle $BAC$ of the triangle $ABC$. For any point $M$ in the plane of the triangle, not on the line $BC$, denote by $I_M$ and $J_M$ the centers of the incircle and the excircle (touching $BC$) of the triangle $BCM$. Find the locus of points $M$ for which $II_MJJ_M$ is a rectangle.

2023 CMWMC, R1

Tags: number theory

[u]Set 1[/u] [b]1.1[/b] How many positive integer divisors are there of $2^2 \cdot 3^3 \cdot 5^4$? [b]1.2[/b] Let $T$ be the answer from the previous problem. For how many integers $n$ between $1$ and $T$ (inclusive) is $\frac{(n)(n - 1)(n - 2)}{12}$ an integer? [b]1.3[/b] Let $T$ be the answer from the previous problem. Find $\frac{lcm(T, 36)}{gcd(T, 36)}$. PS. You should use hide for answers.

2006 Mathematics for Its Sake, 1

Tags: geometry , area , pigeonhole principle , combinatorics

Let be the points $ K,L,M $ on the sides $ BC,CA,AB, $ respectively, of a triangle $ ABC. $ Show that at least one of the areas of the triangles $ MAL,KBM,LCK $ doesn't surpass a fourth of the area of $ ABC. $

2016 Korea - Final Round, 4

Tags: inequality , algebra , inequalities

If $x,y,z$ satisfies $x^2+y^2+z^2=1$, find the maximum possible value of $$(x^2-yz)(y^2-zx)(z^2-xy)$$

1993 AMC 12/AHSME, 10

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Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$ $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 3a^2 \qquad\textbf{(C)}\ 27a^2 \qquad\textbf{(D)}\ 2a^{3b} \qquad\textbf{(E)}\ 3a^{2b} $

2014 Puerto Rico Team Selection Test, 1

Tags: geometry , parallelogram , perpendicular bisector

Let $ABCD$ be a parallelogram with $AB>BC$ and $\angle DAB$ less than $\angle ABC$. The perpendicular bisectors of sides $AB$ and $BC$ intersect at the point $M$ lying on the extension of $AD$. If $\angle MCD=15^{\circ}$, find the measure of $\angle ABC$

2020 JBMO Shortlist, 1

Tags: shortlist , geometry

Let $\triangle ABC$ be an acute triangle. The line through $A$ perpendicular to $BC$ intersects $BC$ at $D$. Let $E$ be the midpoint of $AD$ and $\omega$ the the circle with center $E$ and radius equal to $AE$. The line $BE$ intersects $\omega$ at a point $X$ such that $X$ and $B$ are not on the same side of $AD$ and the line $CE$ intersects $\omega$ at a point $Y$ such that $C$ and $Y$ are not on the same side of $AD$. If both of the intersection points of the circumcircles of $\triangle BDX$ and $\triangle CDY$ lie on the line $AD$, prove that $AB = AC$.

2010 Cuba MO, 8

Tags: geometry , parallelogram , locus

Let $ABCDE$ be a convex pentagon that has $AB < BC$, $AE <ED$ and $AB + CD + EA = BC + DE$. Variable points $F,G$ and $H$ are taken that move on the segments $BC$, $CD$ and $OF$ respectively . $B'$ is defined as the projection of $B$ on $AF$, $C'$ as the projection of $C$ on $FG$, $D'$ as the projection of $D$ on $GH$ and $E'$ as the projection of $E$ onto $HA$. Prove that there is at least one quadrilateral $B'C'D'E'$ when $F,G$ and $H$ move on their sides, which is a parallelogram.

2012 Pre-Preparation Course Examination, 5

Tags: function , calculus , derivative , trigonometry , search , real analysis

The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$. [b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$. [b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.

1996 Singapore Team Selection Test, 2

Tags: algebra , modular arithmetic , number theory

Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.

2020 LMT Fall, A11 B20

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Two sequences of nonzero reals $a_1, a_2, a_3, \dots$ and $b_2, b_3, \dots$ are such that $b_n=\prod_{i=1}^{n} a_i$ and $a_n=\frac{b_n^2}{3b_n-3}$ for all integers $n > 1$. Given that $a_1=\frac{1}{2}$, find $\lvert b_{60}\rvert$. [i]Proposed by Andrew Zhao[/i]

2022 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , combinatorial geometry , combinatorics

Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?

2003 Cuba MO, 2

Tags: geometry , circles , circumcircle , angle

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

2008 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , inequalities

[b]BJ2. [/b] Positive real numbers $ a,b,c$ satisfy inequality $ \frac {3}{2}\geq a \plus{} b \plus{} c$. Find the smallest possible value for $ S \equal{} abc \plus{} \frac {1}{abc}$

2019 HMNT, 8

Tags: algebra

Omkar, Krit1, Krit2, and Krit3 are sharing $x > 0$ pints of soup for dinner. Omkar always takes $1$ pint of soup (unless the amount left is less than one pint, in which case he simply takes all the remaining soup). Krit1 always takes $\frac16$ of what is left, Krit2 always takes $\frac15$ of what is left, and Krit3 always takes $\frac14$ of what is left. They take soup in the order of Omkar, Krit1, Krit2, Krit3, and then cycle through this order until no soup remains. Find all $x$ for which everyone gets the same amount of soup.

2013 JBMO Shortlist, 5

Tags: geometry , geometric inequality

A circle passing through the midpoint $M$ of the side $BC$ and the vertex $A$ of the triangle $ABC$ intersects the segments $AB$ and $AC$ for the second time in the points $P$ and $Q$, respectively. Prove that if $\angle BAC=60^{\circ}$, then $AP+AQ+PQ<AB+AC+\frac{1}{2} BC$.

2010 Indonesia TST, 1

Tags: inequalities

Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $. Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]