How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?

The incircle of triangle $ABC$ meet the sides $AB, AC$ and $BC$ in $M,N$ and $P$, respectively. Prove that the orthocenter of triangle $MNP,$ the incenter and the circumcenter of triangle $ABC$ are collinear. [asy] import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffwwww = rgb(1,0.4,0.4); pen xdxdff = rgb(0.49,0.49,1); draw((8,17.58)--(2.84,9.26)--(20.44,9.21)--cycle); draw((8,17.58)--(2.84,9.26),ttttff+linewidth(2pt)); draw((2.84,9.26)--(20.44,9.21),ttttff+linewidth(2pt)); draw((20.44,9.21)--(8,17.58),ttttff+linewidth(2pt)); draw(circle((9.04,12.66),3.43),blue+linewidth(1.2pt)+linetype("8pt 8pt")); draw((6.04,14.42)--(8.94,9.24),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(11.12,15.48),ffwwww+linewidth(1.2pt)); draw((11.12,15.48)--(6.04,14.42),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(7.81,14.79)); draw((11.12,15.48)--(6.95,12.79)); draw((6.04,14.42)--(10.12,12.6)); dot((8,17.58),ds); label("$A$", (8.11,18.05),NE*lsf); dot((2.84,9.26),ds); label("$B$", (2.11,8.85), NE*lsf); dot((20.44,9.21),ds); label("$C$", (20.56,8.52), NE*lsf); dot((9.04,12.66),ds); label("$O$", (8.94,12.13), NE*lsf); dot((6.04,14.42),ds); label("$M$", (5.32,14.52), NE*lsf); dot((11.12,15.48),ds); label("$N$", (11.4,15.9), NE*lsf); dot((8.94,9.24),ds); label("$P$", (8.91,8.58), NE*lsf); dot((7.81,14.79),ds); label("$D$", (7.81,15.14),NE*lsf); dot((6.95,12.79),ds); label("$F$", (6.64,12.07),NE*lsf); dot((10.12,12.6),ds); label("$G$", (10.41,12.35),NE*lsf); dot((8.07,13.52),ds); label("$H$", (8.11,13.88),NE*lsf); clip((-0.68,-0.96)--(-0.68,25.47)--(30.71,25.47)--(30.71,-0.96)--cycle); [/asy]

We have a deck of $90$ cards that are numbered from $10$ to $99$ (all two-digit numbers). How many sets of three or more different cards in this deck are there such that the number on one of them is the sum of the other numbers, and those other numbers are consecutive?

Let $n\geqslant2$ be a natural number. Show that there is no real number $c{}$ for which \[\exp\left(\frac{T+S}{2}\right)\leqslant c\cdot \frac{\exp(T)+\exp(S)}{2}\]is satisfied for any self-adjoint $n\times n$ complex matrices $T{}$ and $S{}$. (If $A{}$ and $B{}$ are self-adjoint $n\times n$ matrices, $A\leqslant B$ means that $B-A$ is positive semi-definite.)

A rectangular parallelepiped painted blue is cut into $1 \times 1\times 1$ cubes. Find the possible dimensions if the number of cubes without blue faces is equal to one-third of the total number of cubes. [b]Note:[/b] A [i]rectangular parallelepiped[/i] is a solid with $6$ faces, all of which are rectangles (or squares).

Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.

Let $m,n$ be positive integers with $m \geq n$, and let $S$ be the set of all $n$-term sequences of positive integers $(a_1, a_2, \ldots a_n)$ such that $a_1 + a_2 + \cdots + a_n = m$. Show that \[\sum_S 1^{a_1} 2^{a_2} \cdots n^{a_n} = {n \choose n} n^m - {n \choose n-1} (n-1)^m + \cdots + (-1)^{n-2} {n \choose 2} 2^m + (-1)^{n-1} {n \choose 1}.\]

Let $w$ and $h$ be positive integers and define $N(w,h)$ to be the number of ways of arranging $wh$ people of distinct heights for a photoshoot in such a way that they form $w$ columns of $h$ people, with the people of each column sorted by height (i.e. shortest at the front, tallest at the back). Find the largest value of $N(w,h)$ that divides $1008$.

In $xyz$ space, let $l$ be the segment joining two points $(1,\ 0,\ 1)$ and $(1,\ 0,\ 2),$ and $A$ be the figure obtained by revolving $l$ around the $z$ axis. Find the volume of the solid obtained by revolving $A$ around the $x$ axis. Note you may not use double integral.

Let $P$ be a point out of circle $C$. Let $PA$ and $PB$ be the tangents to the circle drawn from $C$. Choose a point $K$ on $AB$ . Suppose that the circumcircle of triangle $PBK$ intersects $C$ again at $T$. Let ${P}'$ be the reflection of $P$ with respect to $A$. Prove that \[ \angle PBT = \angle {P}'KA \]

Let $\triangle{ABC}$ be a triangle, $D$ the midpoint of $BC$, and $M$ be the midpoint of $AD$. The line $BM$ intersects the side $AC$ on the point $N$. Show that $AB$ is tangent to the circuncircle to the triangle $\triangle{NBC}$ if and only if the following equality is true: \[\frac{{BM}}{{MN}} =\frac{({BC})^2}{({BN})^2}.\]

Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized. \[ \int_0^{\pi} |\sin x-ax|\ dx \]

Let $a,b,c$ be positive real numbers. Prove that $\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$

(a) A function $f:\mathbb{Z} \rightarrow \mathbb{Z}$ is called $\mathbb{Z}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \in \mathbb{Z}$. What is the largest possible number of distinct values that can occur among $f(1), \ldots, f(2023)$, where $f$ is a $\mathbb{Z}$-good function? (b) A function $f:\mathbb{N} \rightarrow \mathbb{N}$ is called $\mathbb{N}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \in \mathbb{N}$. What is the largest possible number of distinct values that can occur among $f(1), \ldots, f(2023)$, where $f$ is a $\mathbb{N}$-good function?

$x,y,z$ are positive real numbers such that $xy+yz+zx=1$. prove that: $3-\sqrt{3}+\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge(x+y+z)^2$ (20 points) the exam time was 6 hours.

In $ \vartriangle ABC$, $\angle B = 46^o$ and $\angle C = 48^o$ . A circle is inscribed in $ \vartriangle ABC$ and the points of tangency are connected to form $PQR$. What is the measure of the largest angle in $\vartriangle P QR$?

We will say that a subset $A$ of the set of non-negative integers is $cool$, if there exist an integer $k$, such that for every integer $n\geq k$ there exists exactly one pair of integers $a>b$ from $A$ such that $n=a+b$. Decide, if there exists a $cool$ set.

$n_1<n_2<\ldots<n_k$ are all positive integer numbers $n$, that have the following property: In a square $n \times n$ one can mark $50$ cells so that in any square $3 \times 3$ an odd number of cells are marked. Find $n_{k-2}$

The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

Alice wrote a sequence of $n > 2$ nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those n numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the same as in Alice’s sequence. What are the possible values of $n{}$? [i]Alexey Zaslavsky[/i]

In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.

Let $ABC$ be a scalene triangle with angle bisectors $AD$, $BE$, and $CF$ so that $D$, $E$, and $F$ lie on segments $BC$, $CA$, and $AB$ respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$ respectively. Prove that line $AN$ and the line through $M$ parallel to $AD$ intersect on the circumcircle of $ABC$ if and only if $DE=DF$. [i]Proposed by Michael Ren.[/i]

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

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

[b]Q[/b] On a \(10 \times 10\) chess board whose colors of square are green and blue in an arbitrary way and we are simultaneously allowed to switch all the colors of all squares in any \((2 \times 2)\) and \((5\times 5)\) region. Can we transform any coloring of the board into one where all squares are blue ? Give a proper explanation of your answer. Note. that if a unit square is part of both the $2\times 2$ and $5\times 5$ region,then its color switched is twice(i.e switching is additive) [i]Proposed by Aritra12[/i]